Page 1
TRIGONOMETRY
One mark questions
1. Define an angle.
Solution: An angle is an amount of rotation of a half-line in a
plane about its end from its initial position to a terminal
position.
2. What do you mean by measure of an angle?
Solution: The amount of rotation from initial position to
terminal position is called measure of an angle.
3. Define one radian.
Solution: A Radian is the angle subtended at the centre of a
circle by an arc, whose length is equal to its radius. 1 radian is
denoted by 1𝑐. One radian is also called one circular measure.
4. Convert the following into radian measure:
a) 18° b) 7.5° c) 105° d) 1
2
° e) 36°32′34′′
Solution:
a) 1° = π
180
c ∴ 18° =
π
180× 18
c=
π
10
c
b) 7.5° = π
180× 7.5
c=
π
180×
15
2
c=
π
24
c
c) 105° = π
180× 105
c=
7π
12
c
d) 1
2
°=
π
180×
1
2
c=
π
360
c
e) 36°32′34′′ = 36 +32
60+
34
3600 °
= 36.5427 ° = 36.5427 × 0.0174 c
(since 1° = 0.0174 c) ∴ 36°32′34′′ ≈ 0.6358 radian.
5. Express the following in degree measure:
a) 𝜋
36 b)
7𝜋
6 c)
23𝜋
4 d)
7𝜋
24 e)
5𝜋
12
Page 2
Solution:
a) 1𝑐 = 180
𝜋 ° ∴
𝜋
36=
180
𝜋×
𝜋
36 °
= 5°
b) 7𝜋
6=
180
𝜋×
7𝜋
6 °
= 210°
c) 23𝜋
4=
180
𝜋×
23𝜋
4 °
= 1035°
d) 7𝜋
24=
180
𝜋×
7𝜋
24 °
= 105
2 ∘
= 52.5 °
e) 5𝜋
12=
180
𝜋×
5𝜋
12 °
= 75∘.
6. Prove that 𝑠𝑖𝑛 𝐴 ∙ 𝑐𝑜𝑡 𝐴 = 𝑐𝑜𝑠 𝐴
Solution: 𝐿𝐻𝑆 = 𝑠𝑖𝑛 𝐴 ∙ 𝑐𝑜𝑠 𝐴
𝑠𝑖𝑛 𝐴= 𝑐𝑜𝑠 𝐴 = 𝑅𝐻𝑆.
7. Prove that 𝑐𝑜𝑡 𝐴 ∙ 𝑠𝑒𝑐 𝐴 ∙ sin 𝐴 = 1
Solution: 𝐿𝐻𝑆 = 𝑐𝑜𝑠 𝐴
𝑠𝑖𝑛 𝐴∙
1
cos 𝐴∙ 𝑠𝑖𝑛 𝐴 = 1 = 𝑅𝐻𝑆.
8. Prove that 𝑐𝑜𝑡2 𝜃 ∙ 1 − 𝑐𝑜𝑠2 𝜃 = 𝑐𝑜𝑠2 𝜃
Solution: 𝐿𝐻𝑆 = 𝑐𝑜𝑠 2 𝜃
𝑠𝑖𝑛 2 𝜃∙ 𝑠𝑖𝑛2 𝜃 = 𝑐𝑜𝑠2 𝜃.
9. Prove that 𝑐𝑜𝑠𝑒𝑐 𝛼 ∙ 1 − 𝑠𝑖𝑛2 𝛼 = cot 𝛼
Solution: 𝐿𝐻𝑆 = 1
𝑠𝑖𝑛 𝛼∙ 𝑐𝑜𝑠 𝛼 = 𝑐𝑜𝑡 𝛼 = 𝑅𝐻𝑆.
10. Prove that (1 + 𝑡𝑎𝑛2 𝜃). 1 − 𝑐𝑜𝑠2 𝜃 = 𝑡𝑎𝑛2 𝜃
Solution: 𝐿𝐻𝑆 = 𝑠𝑒𝑐2 𝜃 ∙ 𝑠𝑖𝑛2 𝜃 = 1
𝑐𝑜𝑠 2 𝜃∙ 𝑠𝑖𝑛2 𝜃 = 𝑡𝑎𝑛2 𝜃 = 𝑅𝐻𝑆.
11. Prove that (1 + 𝑐𝑜𝑡2 𝐴) ∙ 𝑠𝑖𝑛2 𝐴 = 1
Solution: 𝐿𝐻𝑆 = 𝑐𝑜𝑠𝑒𝑐2 𝐴 ∙ 𝑠𝑖𝑛2 𝐴
= 𝑐𝑜𝑠𝑒𝑐 𝐴 ∙ 𝑠𝑖𝑛 𝐴 2 = 12 = 1 = 𝑅𝐻𝑆.
Page 3
12. Prove that 𝑐𝑜𝑠 𝛼 ∙ 𝑐𝑜𝑠𝑒𝑐 𝛼 ∙ 𝑠𝑒𝑐2 𝛼 − 1 = 1.
Solution: 𝐿𝐻𝑆 = 𝑐𝑜𝑠 𝛼 ∙1
𝑠𝑖𝑛 𝛼∙ 𝑡𝑎𝑛2 𝛼
= 𝑐𝑜𝑠 𝛼 ∙1
𝑠𝑖𝑛 𝛼∙ 𝑡𝑎𝑛 𝛼 =
𝑐𝑜𝑠 𝛼
𝑠𝑖𝑛 𝛼∙𝑠𝑖𝑛 𝛼
𝑐𝑜𝑠 𝛼 = 1 = 𝑅𝐻𝑆.
13. Prove that 𝑐𝑜𝑠 𝜃 ∙ 𝑐𝑜𝑡2 𝜃 − 1 = 𝑐𝑜𝑠𝑒𝑐2 𝜃 − 1.
Solution: 𝐿𝐻𝑆 = cos 𝜃 ∙ 𝑐𝑜𝑠𝑒𝑐 𝜃
= cos 𝜃 ∙1
𝑠𝑖𝑛 𝜃 = 𝑐𝑜𝑡 𝜃 = 𝑐𝑜𝑠𝑒𝑐2 𝜃 − 1 = 𝑅𝐻𝑆.
14. Prove that 𝑠𝑖𝑛2 𝐴 ∙ 𝑐𝑜𝑡2 𝐴 + 𝑠𝑖𝑛2 𝐴 = 1.
Solution: 𝐿𝐻𝑆 = 𝑠𝑖𝑛2 𝐴 ∙𝑐𝑜𝑠 2 𝐴
𝑠𝑖𝑛 2 𝐴 + 𝑠𝑖𝑛2 𝐴
= 𝑐𝑜𝑠2 𝐴 + 𝑠𝑖𝑛2 𝐴 = 1 = 𝑅𝐻𝑆.
15. Define coterminal angles. Give two examples.
Solution: If the initial and terminal sides of two angles are
equal then the angles are called coterminal angles.
16. Prove that 𝑐𝑜𝑠𝑒𝑐 (180° − 𝜃) = 𝑐𝑜𝑠𝑒𝑐 𝜃.
Solution: 𝑐𝑜𝑠𝑒𝑐 (180° − 𝜃) = 1
𝑠𝑖𝑛 (180° − 𝜃) =
1
𝑠𝑖𝑛 𝜃 = 𝑐𝑜𝑠𝑒𝑐 𝜃.
17. Prove that 𝑠𝑒𝑐 (180° − 𝜃) = − 𝑠𝑒𝑐 𝜃.
Solution: 𝑠𝑒𝑐 (180° − 𝜃) = 1
𝑐𝑜𝑠 (180° − 𝜃) =
1
− 𝑐𝑜𝑠 𝜃 = − 𝑠𝑒𝑐 𝜃.
18. Prove that 𝑐𝑜𝑡 (180° − 𝜃) = − 𝑐𝑜𝑡 𝜃.
Solution:𝑐𝑜𝑡 (180° − 𝜃) = 1
𝑡𝑎𝑛 (180° − 𝜃) =
1
− 𝑡𝑎𝑛 𝜃 = − 𝑐𝑜𝑡 𝜃.
19. Prove that 𝑐𝑜𝑠𝑒𝑐 (180° + 𝜃) = − 𝑐𝑜𝑠𝑒𝑐 𝜃.
Solution: 𝑐𝑜𝑠𝑒𝑐 (180° + 𝜃) = 1
𝑠𝑖𝑛 (180° + 𝜃) =
1
− 𝑠𝑖𝑛 𝜃 = − 𝑐𝑜𝑠𝑒𝑐 𝜃.
20. Prove that 𝑠𝑒𝑐 (180° + 𝜃) = − 𝑠𝑒𝑐 𝜃.
Solution: 𝑠𝑒𝑐 (180° + 𝜃) = 1
𝑐𝑜𝑠 (180° + 𝜃) =
1
− 𝑐𝑜𝑠 𝜃 = − 𝑠𝑒𝑐 𝜃.
Page 4
21. Prove that 𝑐𝑜𝑡 (180° + 𝜃) = − 𝑐𝑜𝑡 𝜃.
Solution: 𝑐𝑜𝑡 (180° + 𝜃) = 1
𝑡𝑎𝑛 (180° + 𝜃) =
1
− 𝑡𝑎𝑛 𝜃 = − 𝑐𝑜𝑡 𝜃.
22. Prove that 𝑐𝑜𝑠𝑒𝑐 (270° − 𝜃) = − 𝑠𝑒𝑐 𝜃.
Solution: 𝑐𝑜𝑠𝑒𝑐 (270° − 𝜃) = 1
𝑠𝑖𝑛 (270° − 𝜃) =
1
− 𝑐𝑜𝑠 𝜃 = − 𝑠𝑒𝑐 𝜃.
23. Prove that 𝑠𝑒𝑐 (270° − 𝜃) = − 𝑐𝑜𝑠𝑒𝑐 𝜃.
Solution: 𝑠𝑒𝑐 (270° − 𝜃) = 1
𝑐𝑜𝑠 (270° − 𝜃) =
1
− 𝑠𝑖𝑛 𝜃 = − 𝑐𝑜𝑠𝑒𝑐 𝜃.
24. Prove that 𝑐𝑜𝑡 (270° − 𝜃) = 𝑡𝑎𝑛 𝜃.
Solution: 𝑐𝑜𝑡 (270° − 𝜃) = 𝑐𝑜𝑡 (180° + 90° − 𝜃)
= 𝑐𝑜𝑡 (90° − 𝜃) = 𝑡𝑎𝑛 𝜃.
25. Prove that 𝑐𝑜𝑠𝑒𝑐 (270° + 𝜃) = − 𝑠𝑒𝑐 𝜃.
Solution: 𝑐𝑜𝑠𝑒𝑐 (270° + 𝜃) = 1
𝑠𝑖𝑛 (270° + 𝜃) =
1
− 𝑐𝑜𝑠 𝜃 = − 𝑠𝑒𝑐 𝜃.
26. Prove that 𝑠𝑒𝑐 (270° + 𝜃) = 𝑐𝑜𝑠𝑒𝑐 𝜃.
Solution: 𝑠𝑒𝑐 (270° + 𝜃) = 1
𝑐𝑜𝑠 (270° + 𝜃) =
1
𝑠𝑖𝑛 𝜃 = 𝑐𝑜𝑠𝑒𝑐 𝜃.
27. Prove that 𝑐𝑜𝑡 (270° + 𝜃) = − 𝑡𝑎𝑛 𝜃.
Solution: 𝑐𝑜𝑡 (270° + 𝜃) = 𝑐𝑜𝑡 (180° + 90° + 𝜃)
= 𝑐𝑜𝑡 (90° + 𝜃) = − 𝑡𝑎𝑛 𝜃.
28. Define periodic function.
Solution: A function 𝑓(𝑥) is said to be periodic if
𝑓( 𝑥 + 𝑇) = 𝑓(𝑥)
for some real number 𝑇.
If 𝑇 is the smallest positive real number satisfying
𝑓( 𝑥 + 𝑇) = 𝑓(𝑥) then 𝑇 is called the period of 𝑓(𝑥).
Page 5
29. Write the period of all the trigonometric functions (1 mark
each).
Solution:
Period of 𝑠𝑖𝑛 𝑥 is 2𝜋 ; Period of 𝑐𝑜𝑠 𝑥 is 2𝜋 ;
Period of 𝑡𝑎𝑛 𝑥 is 𝜋 ; Period of 𝑐𝑜𝑡 𝑥 is 𝜋 ;
Period of 𝑐𝑜𝑠𝑒𝑐 𝑥 is 𝜋 ; Period of 𝑠𝑒𝑐 𝑥 is 𝜋.
30. Prove that 𝑠𝑖𝑛 45° + 𝐴 =1
2(𝑠𝑖𝑛 𝐴 + 𝑐𝑜𝑠 𝐴)
Solution: 𝑠𝑖𝑛 45° + 𝐴 = 𝑠𝑖𝑛 45°. 𝑐𝑜𝑠 𝐴 + 𝑐𝑜𝑠 45°. 𝑠𝑖𝑛 𝐴
= 1
2𝑐𝑜𝑠 𝐴 +
1
2𝑠𝑖𝑛 𝐴) =
1
2(𝑠𝑖𝑛 𝐴 + 𝑐𝑜𝑠 𝐴).
31. Find the value of 𝑠𝑖𝑛 54°
Solution: 𝑠𝑖𝑛 54° = 𝑠𝑖𝑛 (90° − 36°) = 𝑐𝑜𝑠 36° = 5 + 1
4.
32. Find the value of 𝑐𝑜𝑠 72°
Solution: 𝑐𝑜𝑠 72° = 𝑐𝑜𝑠 (90° − 18°) = 𝑠𝑖𝑛 18° = 5 − 1
4.
33. Prove that 2 𝑠𝑖𝑛 15° 𝑐𝑜𝑠15° =1
2.
Solution: 2 𝑠𝑖𝑛 15° 𝑐𝑜𝑠15° = 𝑠𝑖𝑛 2 ∙ 15° = 𝑠𝑖𝑛30° =1
2
34. Prove that 𝑠𝑖𝑛 𝜋
8𝑐𝑜𝑠
𝜋
8=
1
2 2
Solution: 𝐿𝐻𝑆 = 2
2. 𝑠𝑖𝑛
𝜋
8. 𝑐𝑜𝑠
𝜋
8 =
1
2 2𝑠𝑖𝑛
𝜋
8. 𝑐𝑜𝑠
𝜋
8
= 1
2𝑠𝑖𝑛 2 ∙
𝜋
8 =
1
2𝑠𝑖𝑛
𝜋
4 =
1
2×
1
2 =
1
2 2.
35. Prove that 𝑠𝑖𝑛 𝑥
1 + 𝑐𝑜𝑠 𝑥= 𝑡𝑎𝑛
𝑥
2
Solution: 𝑠𝑖𝑛 𝑥
1 + 𝑐𝑜𝑠 𝑥 =
2𝑠𝑖𝑛 𝑥
2.𝑐𝑜𝑠
𝑥
2
2𝑐𝑜𝑠 2 𝑥
2
= 𝑠𝑖𝑛
𝑥
2
𝑐𝑜𝑠 𝑥
2
= 𝑡𝑎𝑛 𝑥
2.
Page 6
36. Prove that 1 + 𝑐𝑜𝑠 2𝑥
1 − 𝑐𝑜𝑠 2𝑥 = 𝑐𝑜𝑡 𝑥
Solution: 1 + 𝑐𝑜𝑠 2𝑥
1 − 𝑐𝑜𝑠 2𝑥 =
2𝑐𝑜𝑠 2 𝑥
2𝑠𝑖𝑛 2 𝑥 =
𝑐𝑜𝑠 𝑥
𝑠𝑖𝑛 𝑥 = 𝑐𝑜𝑡 𝑥.
37. Express 2 𝑠𝑖𝑛2𝐴 ∙ 𝑐𝑜𝑠𝐴 as a sum.
Solution: 2 𝑠𝑖𝑛 2𝐴. 𝑐𝑜𝑠 𝐴 = 𝑠𝑖𝑛 2𝐴 + 𝐴 + 𝑠𝑖𝑛 2𝐴 − 𝐴
= 𝑠𝑖𝑛 3𝐴 + 𝑠𝑖𝑛 𝐴.
38. Express 𝑐𝑜𝑠 7𝜃 ∙ 𝑐𝑜𝑠 5𝜃 as a sum.
Solution: 𝑐𝑜𝑠 7𝜃 ∙ 𝑐𝑜𝑠 5𝜃 = 1
2 𝑐𝑜𝑠 7𝜃 + 5𝜃 + 𝑐𝑜𝑠 7𝜃 − 5𝜃
= 1
2 𝑐𝑜𝑠 12𝜃 + 𝑐𝑜𝑠 2𝜃 .
39. Express 𝑠𝑖𝑛 4𝐴 + 𝑠𝑖𝑛 2𝐴 as product.
Solution: 𝑠𝑖𝑛 4𝐴 + 𝑠𝑖𝑛 2𝐴 = 2 𝑠𝑖𝑛 4𝐴 + 2𝐴
2 . 𝑐𝑜𝑠
4𝐴 − 2𝐴
2
= 2 𝑠𝑖𝑛 3𝐴 . 𝑐𝑜𝑠 𝐴.
40. Express 𝑐𝑜𝑠 60° − 𝑐𝑜𝑠 20° as a product.
Solution: 𝑐𝑜𝑠 60° − 𝑐𝑜𝑠 20° = − 2 𝑠𝑖𝑛 60 ° + 20 °
2 . 𝑠𝑖𝑛
60° − 20°
2
= − 2 𝑠𝑖𝑛 40° ∙ 𝑠𝑖𝑛 20°.
41. Express 𝑐𝑜𝑠 55° + 𝑠𝑖𝑛 55° as a product.
Solution: 𝑐𝑜𝑠 55° + 𝑠𝑖𝑛 55° = 𝑐𝑜𝑠 55° + 𝑐𝑜𝑠 90° − 55°
= 𝑐𝑜𝑠 55° + 𝑐𝑜𝑠 35° = 2 𝑐𝑜𝑠 55 ° − 35 °
2 . 𝑐𝑜𝑠
55° − 35°
2
= 2 𝑐𝑜𝑠 45° . 𝑐𝑜𝑠 10° = 2.1
2. 𝑐𝑜𝑠 10° = 2. 𝑐𝑜𝑠 10°.
42. Define a trigonometric equation.
Solution: An equation involving one or more trigonometric
functions are called trigonometric equation.
Page 7
43. What do you mean by general solution of a trigonometric
equation.
Solution: The set of all values of the angle, 𝜃 (involved in a
trigonometric equation) satisfying the trigonometric equation,
𝑓(𝜃) = 0 is called general solution of the trigonometric
equation.
44. Find the general solution of 𝑠𝑖𝑛 𝜃 = 0.
Solution: When = 0 , ± 𝜋 , ± 2𝜋 , ± 3𝜋, … 𝑠𝑖𝑛 𝜃 = 0.
∴ the general solution of 𝑠𝑖𝑛 𝜃 = 0 is 𝜃 = 𝑛𝜋, 𝑛 ∈ 𝑍.
45. Find the general solution of 𝑐𝑜𝑠 𝜃 = 0.
Solution: When 𝜃 = ± 𝜋
2, ±
3𝜋
2, ±
5𝜋
2 ,… 𝑐𝑜𝑠 𝜃 = 0.
∴ the general solution of 𝑐𝑜𝑠 𝜃 = 0 is 𝜃 = (2𝑛 + 1)𝜋
2 , 𝑛 ∈ 𝑍.
46. Find the general solution of 𝑡𝑎𝑛 𝜃 = 0.
Solution: When 𝜃 = 0 , ± 𝜋 , ± 2𝜋 , ± 3𝜋, … 𝑡𝑎𝑛 𝜃 = 0.
∴ the general solution of 𝑡𝑎𝑛 𝜃 = 0 is 𝜃 = 𝑛𝜋, 𝑛 ∈ 𝑍.
47. Find the principal value of 𝑐𝑜𝑠 𝑥 = 3
2
Solution: 𝑐𝑜𝑠 𝑥 = 3
2> 0 ∴ 𝑥 lies in the I or IV quadrant.
Principal value of 𝑥 ∈ 0, 𝜋 . Since 𝑐𝑜𝑠𝑥 is positive, the principal
value is in the I quadrant.
𝑐𝑜𝑠 𝑥 = 3
2= 𝑐𝑜𝑠
𝜋
6 and
𝜋
6∈ 0, 𝜋 . ∴ principal value of 𝑥 is
𝜋
6.
48. Find the principal value of 𝑠𝑖𝑛 𝑥 =1
2.
Solution: 𝑠𝑖𝑛 𝑥 =1
2> 0 ∴ 𝑥 lies in the I or II quadrant.
Principal value of 𝑥 ∈ −𝜋
2,
𝜋
2 . Since 𝑠𝑖𝑛𝑥 is positive the
principal value is in the I quadrant.
𝑠𝑖𝑛𝑥 =1
2= 𝑠𝑖𝑛
𝜋
6 and
𝜋
6∈ −
𝜋
2,
𝜋
2 . ∴ principal value of 𝑥 is
𝜋
6.
Page 8
49. Find the principal value of 𝑐𝑜𝑠𝑒𝑐 𝜃 = −2
3.
Solution: 𝑐𝑜𝑠𝑒𝑐 𝜃 = − 2
3 ⟹ 𝑠𝑖𝑛 𝜃 = −
3
2< 0
∴ 𝑥 lies in the III or IV quadrant.
Principal value of 𝜃 ∈ −𝜋
2,
𝜋
2 . Since 𝑠𝑖𝑛𝜃 is negative the
principal value is in the IV quadrant.
𝑠𝑖𝑛 𝜃 = − 3
2= 𝑠𝑖𝑛 −
𝜋
3 and −
𝜋
3 ∈ −
𝜋
2,
𝜋
2 .
∴ principal value of 𝑥 is −𝜋
3.
50. Find the principal value of 𝑐𝑜𝑠 𝑥 = − 3
2.
Solution: 𝑐𝑜𝑠 𝑥 = − 3
2< 0 ∴ 𝑥 lies in the II or III quadrant.
Principal value of 𝑥 ∈ 0, 𝜋 . Since 𝑐𝑜𝑠𝑥 is negative the
principal value is in the II quadrant.
𝑐𝑜𝑠𝑥 = − 3
2= − 𝑐𝑜𝑠
𝜋
6= 𝑐𝑜𝑠 180° − 30°
= 𝑐𝑜𝑠150° = 𝑐𝑜𝑠5𝜋
6 and
5𝜋
6∈ 0, 𝜋 .
∴ principal value of 𝑥 𝑖𝑠 5𝜋
6.
Two mark questions
1. Define sexagesimal system.
Solution: In this system, a right angle is divided into 90 equal
parts. Each part is equal to one degree. One degree is divided
into 60 equal parts. Each part is called one minute. One minute
is divided into 60 equal parts. Each part is called one second.
One right angle = 90 degrees, written as 90° ;
one degree, 1° = 60 minutes, written as 60′ ;
one minute, 1′ = 60 seconds, written as 60′′;
Page 9
2. Prove that one radian is a constant angle.
Solution: Since the angle subtended at the center of a circle of
unit radius, by an arc of length one unit is one radian, the
angle subtended at the center by the circumference of the
circle is 2𝜋 radian. Thus 2𝜋𝑐 = 360°.
∴ 𝜋𝑐 = 180°.
∴ 1𝑐 =180
𝜋.
This gives the conclusion, 1𝑐 is a constant angle.
3. With usual notations prove that 𝑙 = 𝑟𝜃.
Solution: We know that in a circle of radius 𝑟, an arc of
length 𝑟, subtends an angle of one radian at the center of the
circle. Since the angle subtended at the center by an arc of
length 𝑟 is one radian, the angle subtended by an arc of length
𝑙 has the measure = 𝑙
𝑟. Thus if 𝜃 is the angle subtended at the
center by an arc of length 𝑙 then 𝑙
𝑟 = 𝜃 ∴ 𝑙 = 𝑟𝜃.
4. With usual notations prove that the area of a sector of a
circle is given by 1
2𝑟2𝜃 or
1
2𝑟𝑙
Solution: We know that in a circle of radius 𝑟, the angle 2𝜋
radian traces the area 𝜋𝑟2. Therefore the area of the sector
tracing an angle 𝜃 radian at the center is 𝐴 =𝜋𝑟2
2𝜋× 𝜃 =
1
2𝑟2𝜃.
Since 𝑙 = 𝑟𝜃 we get 𝐴 =1
2𝑟2𝜃 =
1
2𝑟. 𝑟𝜃 =
1
2𝑟𝑙.
5. An arc of a circle subtends 15° at the centre. If the radius is
4 cm, find the length of the arc and area of the sector formed.
Solution: Here 𝑟 = 4 𝑐𝑚 and 𝜃 = 15° = 15 ×𝜋
180=
𝜋
12
∴ 𝑙 = 𝑟𝜃 ⟹ 𝑙 = 4 ×22
7×
1
12=
22
21 𝑐𝑚.
Area of the sector =1
2𝑟𝑙 =
1
2 × 4 ×
22
21 =
44
21 𝑠𝑞. 𝑐𝑚.
Page 10
6. An arc of length 11
3 cm subtends an angle of 30° at the
centre of a circle. Find the area of the sector of the circle so
formed.
Solution: Given =11
3 , 𝜃 = 30° =
𝜋
6.
We have 𝑙 = 𝑟𝜃 ⟹ 𝑟 =𝑙
𝜃=
11
3×
6
𝜋=
11
3×
6
22× 7 = 7𝑐𝑚
𝐴 =1
2 𝑟2𝜃 =
1
2 7 2 𝜋
6=
1
2 × 49 ×
22
7×
1
6=
77
6 𝑠𝑞. 𝑐𝑚.
7. If, in two circles arcs of the same length subtend angles of
60° 𝑎𝑛𝑑 75° at the centre, find the ratio of their radii.
Solution: Let 𝑟1 and 𝑟2 be the radii of the two circles.
Given 𝜃1 = 60° = 𝜋
180× 60 =
𝜋
3 and 𝜃2 = 75° =
𝜋
180× 75 =
5𝜋
12
Now the length of each of the arc, 𝑙 = 𝑟1𝜃1 = 𝑟2𝜃2, which gives
𝑟1 ×𝜋
3= 𝑟2 ×
5𝜋
12⟹
𝑟1
𝑟2=
5
4. Hence 𝑟1: 𝑟2 = 5: 4.
8. Prove the following:
a) 𝑠𝑖𝑛 𝜃. 𝑐𝑜𝑠𝑒𝑐 𝜃 = 1, 𝑐𝑜𝑠𝑒𝑐 𝜃 = 1
𝑠𝑖𝑛 𝜃 ;
b) 𝑐𝑜𝑠 𝜃. 𝑠𝑒𝑐 𝜃 = 1, 𝑠𝑒𝑐 𝜃 = 1
𝑐𝑜𝑠 𝜃 ;
c) 𝑡𝑎𝑛 𝜃. 𝑐𝑜𝑡 𝜃 = 1, 𝑐𝑜𝑡 𝜃 = 1
𝑡𝑎𝑛 𝜃 ;
d) 𝑠𝑖𝑛 𝜃
𝑐𝑜𝑠 𝜃 = 𝑡𝑎𝑛 𝜃; e)
𝑐𝑜𝑠 𝜃
𝑠𝑖𝑛 𝜃 = 𝑐𝑜𝑡 𝜃.
f) 𝑠𝑖𝑛2𝜃 + 𝑐𝑜𝑠2𝜃 = 1;
g) 1 + 𝑡𝑎𝑛2𝜃 = 𝑠𝑒𝑐2𝜃 ; h) 1 + 𝑐𝑜𝑡2𝜃 = 𝑐𝑜𝑠𝑒𝑐2𝜃.
Solution: Consider a circle of
radius 𝑟 centered at the origin of
the Cartesian coordinate system. Let
𝑃(𝑥, 𝑦) be a point on the circle. Join
𝑂𝑃. Define angle 𝜃 having 𝑂𝑋 has
the initial direction and 𝑂𝑃 as the
terminal direction. Consider a circle
of radius 𝑟 centered at the origin of
Page 11
the Cartesian coordinate system. Let 𝑃(𝑥, 𝑦) be a point on the
circle. Join 𝑂𝑃. Define angle 𝜃 having 𝑂𝑋 has the initial
direction and 𝑂𝑃 as the terminal direction. Then
𝑠𝑖𝑛 𝜃 = 𝑦
𝑟 , 𝑜𝑠 𝜃 =
𝑥
𝑟 , 𝑡𝑎𝑛 𝜃 =
𝑦
𝑥 ,
𝑐𝑜𝑡 𝜃 = 𝑥
𝑦, 𝑠𝑒𝑐 𝜃 =
𝑟
𝑥, 𝑐𝑜𝑠𝑒𝑐 𝜃 =
𝑟
𝑦.
a) 𝑠𝑖𝑛 𝜃. 𝑐𝑜𝑠𝑒𝑐 𝜃 = 𝑦
𝑟.𝑟
𝑦 = 1. ∴ 𝑐𝑜𝑠𝑒𝑐 𝜃 =
1
𝑠𝑖𝑛 𝜃 ;
b) 𝑐𝑜𝑠 𝜃. 𝑠𝑒𝑐 𝜃 = 𝑥
𝑟. 𝑟
𝑥 = 1. ∴ 𝑠𝑒𝑐 𝜃 =
1
𝑐𝑜𝑠 𝜃 ;
c) 𝑡𝑎𝑛 𝜃. 𝑐𝑜𝑡 𝜃 = 𝑦
𝑥.𝑥
𝑦 = 1. ∴ 𝑐𝑜𝑡 𝜃 =
1
𝑡𝑎𝑛 𝜃 ;
d) 𝑠𝑖𝑛 𝜃
𝑐𝑜𝑠 𝜃 =
𝑦 𝑟
𝑥 𝑟 =
𝑦
𝑥 = 𝑡𝑎𝑛 𝜃 ;
e) 𝑐𝑜𝑠 𝜃
𝑠𝑖𝑛 𝜃 =
𝑥 𝑟
𝑦 𝑟 =
𝑥
𝑦 = 𝑐𝑜𝑡 𝜃 ;
f) 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠2 𝜃 = 𝑦
𝑟
2+
𝑥
𝑟
2 =
𝑦2
𝑟2 +
𝑥2
𝑟2 =
𝑦2 + 𝑥2
𝑟2 =
𝑟2
𝑟2 = 1 ;
g) 1 + 𝑡𝑎𝑛2 𝜃 = 1 + 𝑦
𝑥
2 = 1 +
𝑦2
𝑥2 =
𝑥2 + 𝑦2
𝑥2 =
𝑟2
𝑥2 = 𝑠𝑒𝑐2 𝜃 ;
h) 1 + 𝑐𝑜𝑡 2𝜃 = 1 + 𝑥
𝑦
2 = 1 +
𝑥2
𝑦2 =
𝑦 2 + 𝑥2
𝑦2 =
𝑟2
𝑦2 = 𝑐𝑜𝑠𝑒𝑐2 𝜃.
9. Prove that 𝑐𝑜𝑠𝑒𝑐2 𝐴 ∙ 𝑡𝑎𝑛2 𝐴 − 1 = 𝑡𝑎𝑛2 𝐴.
Solution: 𝐿𝐻𝑆 = 1 + 𝑐𝑜𝑡2𝐴 ∙ 𝑡𝑎𝑛2 𝐴 − 1
= 𝑡𝑎𝑛2 𝐴 + 𝑐𝑜𝑡2 𝐴 ∙ 𝑡𝑎𝑛2 𝐴 − 1
= 𝑡𝑎𝑛2 𝐴 + 𝑐𝑜𝑡 𝐴 ∙ 𝑡𝑎𝑛 𝐴 2 − 1
= 𝑡𝑎𝑛2 𝐴 + 1 − 1 = 𝑡𝑎𝑛2 𝐴 = 𝑅𝐻𝑆.
10. Prove that 𝑐𝑜𝑠 𝛼
𝑠𝑒𝑐 𝛼+
𝑠𝑖𝑛 𝛼
𝑐𝑜𝑠𝑒𝑐 𝛼= 1.
Solution: 𝐿𝐻𝑆 = 𝑐𝑜𝑠 𝛼
1
𝑐𝑜𝑠 𝛼
+ 𝑠𝑖𝑛 𝛼
1
𝑠𝑖𝑛 𝛼
= 𝑐𝑜𝑠2 𝛼 + 𝑠𝑖𝑛2 𝛼 = 1 = 𝑅𝐻𝑆.
11. Prove that 𝑐𝑜𝑠4 𝐴 − 𝑠𝑖𝑛4 𝐴 = 2 𝑐𝑜𝑠2 𝐴 − 1.
Solution: 𝐿𝐻𝑆 = 𝑐𝑜𝑠2 𝐴 2 − 𝑠𝑖𝑛2 𝐴 2
= 𝑐𝑜𝑠2 𝐴 + 𝑠𝑖𝑛2 𝐴 𝑐𝑜𝑠2 𝐴 − 𝑠𝑖𝑛2 𝐴
= 1 ∙ 𝑐𝑜𝑠2 𝐴 − 𝑠𝑖𝑛2 𝐴
Page 12
= 𝑐𝑜𝑠2 𝐴 − 1 − 𝑐𝑜𝑠2 𝐴
= 𝑐𝑜𝑠2 𝐴 − 1 + 𝑐𝑜𝑠2 𝐴 = 2𝑐𝑜𝑠2 𝐴 − 1 = 𝑅𝐻𝑆 .
12. Prove that 𝑠𝑒𝑐 𝜃 ∙ 𝑐𝑜𝑡 𝜃 2 − 𝑐𝑜𝑠 𝜃 ∙ 𝑐𝑜𝑠𝑒𝑐 𝜃 2 = 1.
Solution: 𝐿𝐻𝑆 = 1
cos 𝜃∙
cos 𝜃
sin 𝜃
2− cos 𝜃 ∙
1
sin 𝜃
2
= 1
sin 𝜃
2−
cos 𝜃
sin 𝜃
2 = cosec 𝜃 2 − cot 𝜃 2 = 1 = 𝑅𝐻𝑆.
13. Prove that 𝑠𝑒𝑐 𝜃 − 𝑡𝑎𝑛 𝜃 ∙ 𝑠𝑖𝑛 𝜃 = 𝑐𝑜𝑠 𝜃.
Solution: 𝐿𝐻𝑆 = 1
𝑐𝑜𝑠 𝜃 −
𝑠𝑖𝑛 𝜃
𝑐𝑜𝑠 𝜃∙ 𝑠𝑖𝑛 𝜃
= 1
𝑐𝑜𝑠 𝜃−
𝑠𝑖𝑛 2 𝜃
𝑐𝑜𝑠 𝜃=
1−𝑠𝑖𝑛 2 𝜃
𝑐𝑜𝑠 𝜃 =
𝑐𝑜𝑠 𝜃 2
𝑐𝑜𝑠 𝜃= 𝑐𝑜𝑠 𝜃 = 𝑅𝐻𝑆.
14. Prove that 𝑡𝑎𝑛 𝜃 + 𝑐𝑜𝑡 𝜃 = 𝑠𝑒𝑐 𝜃 ∙ 𝑐𝑜𝑠𝑒𝑐 𝜃.
Solution: 𝐿𝐻𝑆 = 𝑠𝑖𝑛 𝜃
𝑐𝑜𝑠 𝜃 +
𝑐𝑜𝑠 𝜃
𝑠𝑖𝑛 𝜃
= 𝑠𝑖𝑛 2 𝜃 + 𝑐𝑜𝑠 2 𝜃
𝑐𝑜𝑠 𝜃∙ 𝑠𝑖𝑛 𝜃
= 1
𝑐𝑜𝑠 𝜃 ∙
1
𝑠𝑖𝑛 𝜃 = 𝑠𝑒𝑐 𝜃 ∙ 𝑐𝑜𝑠𝑒𝑐 𝜃 = 𝑅𝐻𝑆.
15. Prove that 𝑐𝑜𝑠 𝜃 + 𝑠𝑖𝑛 𝜃 2 + 𝑐𝑜𝑠 𝜃 − 𝑠𝑖𝑛 𝜃 2 = 2.
Solution:
𝐿𝐻𝑆 = 𝑐𝑜𝑠2 𝜃 + 𝑠𝑖𝑛2 𝜃 + 2 𝑐𝑜𝑠 𝜃 . 𝑠𝑖𝑛 𝜃
+ 𝑐𝑜𝑠2 𝜃 + 𝑠𝑖𝑛2 𝜃 − 2 cos 𝜃 . sin 𝜃 = 1 + 1 = 2 = 𝑅𝐻𝑆.
16. Prove that 𝑡𝑎𝑛2𝛼 + 𝑡𝑎𝑛4𝛼 = 𝑠𝑒𝑐4𝛼 − 𝑠𝑒𝑐2𝛼.
Solution: 𝐿𝐻𝑆 = 𝑡𝑎𝑛2𝛼 + 𝑡𝑎𝑛4𝛼 = 𝑡𝑎𝑛2𝛼 1 + 𝑡𝑎𝑛2𝛼
= 𝑠𝑒𝑐2𝛼 − 1 𝑠𝑒𝑐2𝛼 = 𝑠𝑒𝑐4𝛼 − 𝑠𝑒𝑐2𝛼 = 𝑅𝐻𝑆.
17. Prove that 1
1 − 𝑐𝑜𝑠 𝛼+
1
1 + 𝑐𝑜𝑠 𝛼= 2 𝑐𝑜𝑠𝑒𝑐2𝛼
Solution: 𝐿𝐻𝑆 = 1 + 𝑐𝑜𝑠 𝛼 + 1 − 𝑐𝑜𝑠 𝛼
1 − 𝑐𝑜𝑠 𝛼 1 − 𝑐𝑜𝑠 𝛼
=2
1 − 𝑐𝑜𝑠 2 𝛼 =
2
𝑠𝑖𝑛 2 𝛼 = 2 𝑐𝑜𝑠𝑒𝑐2𝛼 = 𝑅𝐻𝑆.
18. Prove that 1
1 + 𝑐𝑜𝑠 2 𝛼+
1
1 + 𝑠𝑒𝑐 2 𝛼 = 1.
Page 13
Solution: 𝐿𝐻𝑆 = 1
1 + 𝑐𝑜𝑠 2 𝛼 +
1
1 + 1
𝑐𝑜𝑠 2 𝛼
=1
1+ 𝑐𝑜𝑠 2 𝛼 +
𝑐𝑜𝑠 2 𝛼
1+ 𝑐𝑜𝑠 2 𝛼 =
1 + 𝑐𝑜𝑠 2 𝛼
1 + 𝑐𝑜𝑠 2 𝛼= 1 = 𝑅𝐻𝑆.
19. Prove that 𝑠𝑖𝑛290° + 𝑐𝑜𝑠260° − 2 𝑠𝑖𝑛245° +3
2𝑠𝑖𝑛230° =
5
8.
Solution: 𝐿𝐻𝑆 = 12 + 1
2
2− 2
1
2
2+
3
2
1
2
2
= 1 + 1
4 − 1 +
3
8 =
1
4 +
3
8 =
5
8 = 𝑅𝐻𝑆.
20. Prove that 𝑐𝑜𝑡2 𝜋
6 − 2𝑐𝑜𝑠2 𝜋
3 −
3
4𝑠𝑒𝑐2 𝜋
4 − 4𝑠𝑖𝑛2 𝜋
6 = 0.
Solution: 𝐿𝐻𝑆 = 3 2− 2
1
2
2−
3
4 2
2+ 4
1
2
2
= 3 − 1
2 −
3
2 − 1 = 0 = 𝑅𝐻𝑆.
21. Prove that 𝑐𝑜𝑠𝑒𝑐245° ∙ 𝑐𝑜𝑠60° ∙ sin90° ∙ 𝑠𝑒𝑐230° =4
3.
Solution: 𝐿𝐻𝑆 = 2 2∙
1
2∙ 1 ∙
2
3
2= 2 ∙
1
2∙ 1 ∙
4
3 =
4
3 = 𝑅𝐻𝑆.
22. Prove that 𝑡𝑎𝑛260° + 2 𝑡𝑎𝑛245° = 5.
Solution: 𝐿𝐻𝑆 = 3 2
+ 2 = 3 + 2 = 5 = 𝑅𝐻𝑆.
23. Prove that 𝑡𝑎𝑛2 𝜋
6+ 𝑡𝑎𝑛2 𝜋
4+ 𝑡𝑎𝑛2 𝜋
3=
13
3 .
Solution: 𝐿𝐻𝑆 = 1
3
2+ 1 2 + 3
2
= 1
3 + 1 + 3 =
1 + 3 + 9
3 =
13
3= 𝑅𝐻𝑆.
24. Prove that 1 − 𝑐𝑜𝑠30°
1 + 𝑐𝑜𝑠30° =
2 − 3
2 + 3
Solution: 𝐿𝐻𝑆 =1 −
3
2
1 + 3
2
= 2 − 3
2
2 + 3
2
=2 − 3
2 + 3= 𝑅𝐻𝑆.
Page 14
25. Find 𝑥, 𝑥 𝑠𝑖𝑛2 𝜋
4 . 𝑐𝑜𝑠2 𝜋
4= 𝑡𝑎𝑛2 𝜋
3
Solution: 𝑥 1
2
2∙
1
2
2= 3
2 ⟹
𝑥
4= 3 ∴ 𝑥 = 12.
26. Prove that trigonometric ratios of 2𝜋𝑛 + 𝜃 are same as the
trigonometric ratio of 𝜃.
Solution: 𝜃, 2𝜋 + 𝜃, 4𝜋 + 𝜃, ... are co-terminal angles. In
general 𝜃 is co-terminal with the angle 2𝜋𝑛 + 𝜃, ∀ 𝑛 ∈ 𝑍.
Since the points corresponding to the coterminal angles are the
same, the trigonometric ratios of coterminal angles are the
same. Thus 𝑠𝑖𝑛 (2𝜋𝑛 + 𝜃) = 𝑠𝑖𝑛 𝜃, 𝑐𝑜𝑠 (2𝜋𝑛 + 𝜃) = 𝑐𝑜𝑠 𝜃 and
𝑡𝑎𝑛 (2𝜋𝑛 + 𝜃) = 𝑡𝑎𝑛 𝜃, ∀ 𝑛 ∈ 𝑍.
27. Prove that trigonometric ratios of 2𝜋𝑛 − 𝜃, ∀ 𝑛 ∈ 𝑍 are
same as the trigonometric ratio of − 𝜃.
Solution: − 𝜃, 2𝜋 − 𝜃, 4𝜋 − 𝜃, ... are co-terminal angles. In
general − 𝜃 is co-terminal with the angle 2𝜋𝑛 − 𝜃, ∀ 𝑛 ∈ 𝑍.
Since the points corresponding to the coterminal angles are the
same, the trigonometric ratios of coterminal angles are the
same. Thus 𝑠𝑖𝑛 (2𝜋𝑛 − 𝜃) = 𝑠𝑖𝑛 (− 𝜃), 𝑐𝑜𝑠 (2𝜋𝑛 − 𝜃) = 𝑐𝑜𝑠 ( − 𝜃)
and 𝑡𝑎𝑛 (2𝜋𝑛 − 𝜃) = 𝑡𝑎𝑛 ( − 𝜃), ∀ 𝑛 ∈ 𝑍.
28. Prove that 𝑠𝑖𝑛 (180° − 𝜃) = 𝑠𝑖𝑛 𝜃.
Solution: 𝑠𝑖𝑛 (180° − 𝜃) = 𝑠𝑖𝑛 180°. 𝑐𝑜𝑠 𝜃 − 𝑐𝑜𝑠 180°. 𝑠𝑖𝑛 𝜃
= 0. 𝑐𝑜𝑠 𝜃 − (− 1). 𝑠𝑖𝑛 𝜃 = 𝑠𝑖𝑛 𝜃.
29. Prove that 𝑐𝑜𝑠 (180° − 𝜃) = − 𝑐𝑜𝑠 𝜃.
Solution: 𝑐𝑜𝑠 (180° − 𝜃) = 𝑐𝑜𝑠 180°. 𝑐𝑜𝑠 𝜃 + 𝑠𝑖𝑛 180°. 𝑠𝑖𝑛 𝜃
= (− 1). 𝑐𝑜𝑠 𝜃 + 0. 𝑠𝑖𝑛 𝜃 = − 𝑐𝑜𝑠 𝜃.
30. Prove that 𝑡𝑎𝑛 (180° − 𝜃) = − 𝑡𝑎𝑛 𝜃.
Solution: 𝑡𝑎𝑛 (180° − 𝜃) = 𝑡𝑎𝑛 180° − 𝑡𝑎𝑛 𝜃
1 + 𝑡𝑎𝑛 180°.𝑡𝑎𝑛 𝜃
= 0 − 𝑡𝑎𝑛 𝜃
1 + 0.𝑡𝑎𝑛 𝜃 = − 𝑡𝑎𝑛 𝜃.
Page 15
31. Prove that 𝑠𝑖𝑛 (180° + 𝜃) = − 𝑠𝑖𝑛 𝜃.
Solution: 𝑠𝑖𝑛 (180° + 𝜃) = 𝑠𝑖𝑛 180°. 𝑐𝑜𝑠 𝜃 + 𝑐𝑜𝑠 180°. 𝑠𝑖𝑛 𝜃
= 0. 𝑐𝑜𝑠 𝜃 + (− 1). 𝑠𝑖𝑛 𝜃 = − 𝑠𝑖𝑛 𝜃.
32. Prove that 𝑐𝑜𝑠 (180° + 𝜃) = − 𝑐𝑜𝑠 𝜃.
Solution: 𝑐𝑜𝑠 (180° + 𝜃) = 𝑐𝑜𝑠 180°. 𝑐𝑜𝑠 𝜃 − 𝑠𝑖𝑛 180°. 𝑠𝑖𝑛 𝜃
= (− 1). 𝑐𝑜𝑠 𝜃 − 0. 𝑠𝑖𝑛 𝜃 = − 𝑐𝑜𝑠 𝜃.
33. Prove that 𝑡𝑎𝑛 (180° + 𝜃) = 𝑡𝑎𝑛 𝜃.
Solution: 𝑡𝑎𝑛 (180° + 𝜃) = 𝑡𝑎𝑛 180° + 𝑡𝑎𝑛 𝜃
1 − 𝑡𝑎𝑛 180°.𝑡𝑎𝑛 𝜃
= 0 + 𝑡𝑎𝑛 𝜃
1 − 0.𝑡𝑎𝑛 𝜃 = 𝑡𝑎𝑛 𝜃.
34. Prove that 𝑠𝑖𝑛 (270° − 𝜃) = − 𝑐𝑜𝑠 𝜃.
Solution: 𝑠𝑖𝑛 (270° − 𝜃) = 𝑠𝑖𝑛 270°. 𝑐𝑜𝑠 𝜃 − 𝑐𝑜𝑠 270°. 𝑠𝑖𝑛 𝜃
= − 1. 𝑐𝑜𝑠 𝜃 − 0. 𝑠𝑖𝑛 𝜃 = − 𝑐𝑜𝑠 𝜃.
35. Prove that 𝑐𝑜𝑠 (270° − 𝜃) = 𝑠𝑖𝑛 𝜃.
Solution: 𝑐𝑜𝑠 (270° − 𝜃) = 𝑐𝑜𝑠 270°. 𝑐𝑜𝑠 𝜃 + 𝑠𝑖𝑛 270°. 𝑠𝑖𝑛 𝜃
= 0. 𝑐𝑜𝑠 𝜃 + (− 1). 𝑠𝑖𝑛 𝜃 = − 𝑠𝑖𝑛 𝜃.
36. Prove that 𝑡𝑎𝑛 (270° − 𝜃) = 𝑐𝑜𝑡 𝜃.
Solution: 𝑡𝑎𝑛 (270° − 𝜃) = 𝑡𝑎𝑛 (180° + 90° − 𝜃)
= 𝑡𝑎𝑛 ( 90° − 𝜃) = 𝑐𝑜𝑡 𝜃.
37. Prove that 𝑠𝑖𝑛 (270° + 𝜃) = − 𝑐𝑜𝑠 𝜃.
Solution: 𝑠𝑖𝑛 (270° + 𝜃) = 𝑠𝑖𝑛 270°. 𝑐𝑜𝑠 𝜃 + 𝑐𝑜𝑠 270°. 𝑠𝑖𝑛 𝜃
= − 1. 𝑐𝑜𝑠 𝜃 + 0. 𝑠𝑖𝑛 𝜃 = − 𝑐𝑜𝑠 𝜃.
38. Prove that 𝑐𝑜𝑠 (270° + 𝜃) = 𝑠𝑖𝑛 𝜃.
Solution: 𝑐𝑜𝑠 (270° + 𝜃) = 𝑐𝑜𝑠 270°. 𝑐𝑜𝑠 𝜃 − 𝑠𝑖𝑛 270°. 𝑠𝑖𝑛 𝜃
= 0. 𝑐𝑜𝑠 𝜃 − (− 1). 𝑠𝑖𝑛 𝜃 = 𝑠𝑖𝑛 𝜃.
39. Prove that 𝑡𝑎𝑛 (270° + 𝜃) = − 𝑐𝑜𝑡 𝜃.
Page 16
Solution: 𝑡𝑎𝑛 (270° + 𝜃) = 𝑡𝑎𝑛 (180° + 90° + 𝜃)
= 𝑡𝑎𝑛 ( 90° + 𝜃) = − 𝑐𝑜𝑡 𝜃.
40. Prove that 𝑡𝑎𝑛 (𝑥 + 𝑦) = 𝑡𝑎𝑛 𝑥 + 𝑡𝑎𝑛 𝑦
1 − 𝑡𝑎𝑛 𝑥 .𝑡𝑎𝑛 𝑦
Solution: 𝑡𝑎𝑛 (𝑥 + 𝑦) = 𝑠𝑖𝑛 (𝑥 + 𝑦)
𝑐𝑜𝑠 (𝑥 + 𝑦) =
𝑠𝑖𝑛 𝑥 .𝑐𝑜𝑠 𝑦 + 𝑐𝑜𝑠 𝑥 .𝑠𝑖𝑛 𝑦
𝑐𝑜𝑠 𝑥 .𝑐𝑜𝑠 𝑦 − 𝑠𝑖𝑛 𝑥 .𝑠𝑖𝑛 𝑦.
Dividing the numerator and the denominator by 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 we
get, 𝑡𝑎𝑛 (𝑥 + 𝑦) =
𝑠𝑖𝑛 𝑥 .𝑐𝑜𝑠 𝑦
𝑐𝑜𝑠 𝑥 .𝑐𝑜𝑠 𝑦 +
𝑐𝑜𝑠 𝑥 .𝑠𝑖𝑛 𝑦
𝑐𝑜𝑠 𝑥 .𝑐𝑜𝑠 𝑦
𝑐𝑜𝑠 𝑥 .𝑐𝑜𝑠 𝑦
𝑐𝑜𝑠 𝑥 .𝑐𝑜𝑠 𝑦 −
𝑠𝑖𝑛 𝑥 .𝑠𝑖𝑛 𝑦
𝑐𝑜𝑠 𝑥 .𝑐𝑜𝑠 𝑦
= 𝑡𝑎𝑛 𝑥 + 𝑡𝑎𝑛 𝑦
1 − 𝑡𝑎𝑛 𝑥 .𝑡𝑎𝑛 𝑦 .
41. Prove that 𝑡𝑎𝑛 (𝑥 − 𝑦) = 𝑡𝑎𝑛 𝑥 − 𝑡𝑎𝑛 𝑦
1 + 𝑡𝑎𝑛 𝑥 .𝑡𝑎𝑛 𝑦
Solution: 𝑡𝑎𝑛 (𝑥 − 𝑦) = 𝑠𝑖𝑛 (𝑥 − 𝑦)
𝑐𝑜𝑠 (𝑥 − 𝑦) =
𝑠𝑖𝑛 𝑥 .𝑐𝑜𝑠 𝑦 − 𝑐𝑜𝑠 𝑥 .𝑠𝑖𝑛 𝑦
𝑐𝑜𝑠 𝑥 .𝑐𝑜𝑠 𝑦 + 𝑠𝑖𝑛 𝑥 .𝑠𝑖𝑛 𝑦.
Dividing the numerator and the denominator by 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 we
get, 𝑡𝑎𝑛 (𝑥 − 𝑦) =
𝑠𝑖𝑛 𝑥 .𝑐𝑜𝑠 𝑦
𝑐𝑜𝑠 𝑥 .𝑐𝑜𝑠 𝑦 −
𝑐𝑜𝑠 𝑥 .𝑠𝑖𝑛 𝑦
𝑐𝑜𝑠 𝑥 .𝑐𝑜𝑠 𝑦
𝑐𝑜𝑠 𝑥 .𝑐𝑜𝑠 𝑦
𝑐𝑜𝑠 𝑥 .𝑐𝑜𝑠 𝑦 +
𝑠𝑖𝑛 𝑥 .𝑠𝑖𝑛 𝑦
𝑐𝑜𝑠 𝑥 .𝑐𝑜𝑠 𝑦
= 𝑡𝑎𝑛 𝑥 − 𝑡𝑎𝑛 𝑦
1 + 𝑡𝑎𝑛 𝑥 .𝑡𝑎𝑛 𝑦 .
42. Prove that 𝑐𝑜𝑡 (𝑥 + 𝑦) = 𝑐𝑜𝑡 𝑥 .𝑐𝑜𝑡 𝑦 − 1
𝑐𝑜𝑡 𝑥 + 𝑐𝑜𝑡 𝑦
Solution: We know that 𝑡𝑎𝑛 (𝑥 + 𝑦) =𝑡𝑎𝑛 𝑥 + 𝑡𝑎𝑛 𝑦
1 − 𝑡𝑎𝑛 𝑥 .𝑡𝑎𝑛 𝑦 .
∴ 𝑐𝑜𝑡 (𝑥 + 𝑦) =1 − 𝑡𝑎𝑛 𝑥 .𝑡𝑎𝑛 𝑦
𝑡𝑎𝑛 𝑥 + 𝑡𝑎𝑛 𝑦
=1 −
1
𝑐𝑜𝑡 𝑥 .𝑐𝑜𝑡 𝑦1
𝑐𝑜𝑡 𝑥 +
1
𝑐𝑜𝑡 𝑦
=
𝑐𝑜𝑡 𝑥 .𝑐𝑜𝑡 𝑦 − 1
𝑐𝑜𝑡 𝑥 .𝑐𝑜𝑡 𝑦𝑐𝑜𝑡 𝑦 + 𝑐𝑜𝑡 𝑥
𝑐𝑜𝑡 𝑥 .𝑐𝑜𝑡 𝑦
= 𝑐𝑜𝑡 𝑥 .𝑐𝑜𝑡 𝑦 − 1
𝑐𝑜𝑡 𝑥 + 𝑐𝑜𝑡 𝑦
43. Prove that 𝑐𝑜𝑡 (𝑥 − 𝑦) = 𝑐𝑜𝑡 𝑥 .𝑐𝑜𝑡 𝑦 + 1
𝑐𝑜𝑡 𝑦 − 𝑐𝑜𝑡 𝑥.
Solution: We know that 𝑡𝑎𝑛 (𝑥 − 𝑦) =𝑡𝑎𝑛 𝑥 − 𝑡𝑎𝑛 𝑦
1 + 𝑡𝑎𝑛 𝑥 .𝑡𝑎𝑛 𝑦 .
∴ 𝑐𝑜𝑡 (𝑥 − 𝑦) =1 + 𝑡𝑎𝑛 𝑥 .𝑡𝑎𝑛 𝑦
𝑡𝑎𝑛 𝑥 − 𝑡𝑎𝑛 𝑦
= 1 +
1
𝑐𝑜𝑡 𝑥 .𝑐𝑜𝑡 𝑦1
𝑐𝑜𝑡 𝑥 −
1
𝑐𝑜𝑡 𝑦
=
𝑐𝑜𝑡 𝑥 .𝑐𝑜𝑡 𝑦 + 1
𝑐𝑜𝑡 𝑥 .𝑐𝑜𝑡 𝑦𝑐𝑜𝑡 𝑦 − 𝑐𝑜𝑡 𝑥
𝑐𝑜𝑡 𝑥 .𝑐𝑜𝑡 𝑦
=𝑐𝑜𝑡 𝑥 .𝑐𝑜𝑡 𝑦 + 1
𝑐𝑜𝑡 𝑦 − 𝑐𝑜𝑡 𝑥.
44. Prove that 𝑠𝑖𝑛 (𝑥 + 𝑦). 𝑠𝑖𝑛 (𝑥 − 𝑦) = 𝑠𝑖𝑛2 𝑥 − 𝑠𝑖𝑛2 𝑦
Solution: 𝑠𝑖𝑛 (𝑥 + 𝑦). 𝑠𝑖𝑛 (𝑥 − 𝑦)
= (𝑠𝑖𝑛 𝑥. 𝑐𝑜𝑠 𝑦 + 𝑐𝑜𝑠 𝑥. 𝑠𝑖𝑛 𝑦). (𝑠𝑖𝑛 𝑥. 𝑐𝑜𝑠 𝑦 − 𝑐𝑜𝑠 𝑥. 𝑠𝑖𝑛 𝑦)
Page 17
= 𝑠𝑖𝑛2 𝑥. 𝑐𝑜𝑠2𝑦 − 𝑐𝑜𝑠2 𝑥. 𝑠𝑖𝑛2 𝑦
= 𝑠𝑖𝑛2 𝑥(1 − 𝑠𝑖𝑛2𝑦) − (1 − 𝑠𝑖𝑛2 𝑥)𝑠𝑖𝑛2 𝑦
= 𝑠𝑖𝑛2 𝑥 − 𝑠𝑖𝑛2 𝑥. 𝑠𝑖𝑛2 𝑦 − 𝑠𝑖𝑛2 𝑦 + 𝑠𝑖𝑛2 𝑥. 𝑠𝑖𝑛2 𝑦
= 𝑠𝑖𝑛2 𝑥 − 𝑠𝑖𝑛2 𝑦.
45. Prove that 𝑐𝑜𝑠 (𝑥 + 𝑦). 𝑐𝑜𝑠 (𝑥 − 𝑦) = 𝑐𝑜𝑠2 𝑥 − 𝑠𝑖𝑛2 𝑦
Solution: 𝑐𝑜𝑠 (𝑥 + 𝑦). 𝑐𝑜𝑠 (𝑥 − 𝑦)
= (𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 − 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦).(𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 + 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦)
= 𝑐𝑜𝑠2 𝑥. 𝑐𝑜𝑠2𝑦 − 𝑠𝑖𝑛2 𝑥. 𝑠𝑖𝑛2 𝑦
= 𝑐𝑜𝑠2 𝑥(1 − 𝑠𝑖𝑛2𝑦) − (1 − 𝑐𝑜𝑠2 𝑥)𝑠𝑖𝑛2 𝑦
= 𝑐𝑜𝑠2 𝑥 − 𝑐𝑜𝑠2 𝑥. 𝑠𝑖𝑛2 𝑦 − 𝑠𝑖𝑛2 𝑦 + 𝑐𝑜𝑠2 𝑥. 𝑠𝑖𝑛2 𝑦
= 𝑐𝑜𝑠2 𝑥 − 𝑠𝑖𝑛2 𝑦.
46. Find the values of the following
a) 𝑠𝑖𝑛 15° ; b) 𝑐𝑜𝑠 15° ; c) 𝑡𝑎𝑛 15° ;
d) 𝑠𝑖𝑛 75° ; e) 𝑐𝑜𝑠 75° ; f) 𝑡𝑎𝑛 75°.
Solution: a) 𝑠𝑖𝑛 15° = 𝑠𝑖𝑛 (45° − 30°)
= 𝑠𝑖𝑛 45°. 𝑐𝑜𝑠 30° − 𝑐𝑜𝑠 45°. 𝑠𝑖𝑛 30°
= 1
2. 3
2 −
1
2.
1
2 =
3 − 1
2 2.
b) 𝑐𝑜𝑠 15° = 𝑐𝑜𝑠 (45° − 30°)
= 𝑐𝑜𝑠 45°. 𝑐𝑜𝑠 30° + 𝑠𝑖𝑛 45°. 𝑠𝑖𝑛 30°
= 1
2. 3
2 +
1
2.
1
2 =
3 + 1
2 2
c) 𝑡𝑎𝑛 15° = 𝑡𝑎𝑛 (45° − 30°)
= 𝑡𝑎𝑛 45° − 𝑡𝑎𝑛 30°
1 + 𝑡𝑎𝑛 45°.𝑡𝑎𝑛 30° =
1 − 1
3
1 + 1.1
3
= 3 − 1
3 + 1
= 3 − 1
3 + 1. 3 − 1
3 − 1 =
3 − 1 2
32
− 1 =
3 + 1 − 2 3
2
= 4 − 2 3
2 = 2 − 3.
d) 𝑠𝑖𝑛 75° = 𝑠𝑖𝑛 (45° + 30°)
= 𝑠𝑖𝑛 45°. 𝑐𝑜𝑠 30° + 𝑐𝑜𝑠 45°. 𝑠𝑖𝑛 30°
= 1
2. 3
2 +
1
2.
1
2 =
3 + 1
2 2.
e) 𝑐𝑜𝑠 75° = 𝑐𝑜𝑠 (45° + 30°)
Page 18
= 𝑐𝑜𝑠 45°. 𝑐𝑜𝑠 30° − 𝑠𝑖𝑛 45°. 𝑠𝑖𝑛 30°
= 1
2. 3
2 −
1
2.
1
2 =
3 − 1
2 2
f) 𝑡𝑎𝑛 75° = 𝑡𝑎𝑛 (45° + 30°)
= 𝑡𝑎𝑛 45° + 𝑡𝑎𝑛 30°
1 − 𝑡𝑎𝑛 45°.𝑡𝑎𝑛 30° =
1 + 1
3
1 − 1.1
3
= 3 + 1
3 − 1
= 3 + 1
3 − 1. 3 + 1
3 + 1 =
3 + 1 2
32
− 1 =
3 + 1 + 2 3
2
= 4 + 2 3
2 = 2 + 3.
47. Find the values of the following
a) 𝑐𝑜𝑠𝑒𝑐 15° ; b) 𝑠𝑒𝑐 15° ; c) 𝑐𝑜𝑡 15°;
d) 𝑐𝑜𝑠𝑒𝑐 75° ; e) 𝑠𝑒𝑐 75° ; f) 𝑐𝑜𝑡 75°.
Solution: a) 𝑐𝑜𝑠𝑒𝑐 15° = 1
𝑠𝑖𝑛 15° =
2 2
3 − 1 =
2 2
3 − 1. 3 + 1
3 + 1
= 2 2( 3 + 1)
3 − 1 = 2( 3 + 1) = 6 + 2.
b) 𝑠𝑒𝑐 15° = 1
𝑐𝑜𝑠 15° =
2 2
3 + 1 =
2 2
3 + 1. 3 − 1
3 − 1 =
2 2( 3 − 1)
3 − 1
= 2( 3 − 1) = 6 − 2.
c) 𝑐𝑜𝑡 15° = 1
𝑡𝑎𝑛 15° =
1
2 − 3 =
1
2 − 3.
2 + 3
2 + 3 =
2 + 3
4 − 3 = 2 + 3.
d) 𝑐𝑜𝑠𝑒𝑐 75° = 1
𝑠𝑖𝑛 75° =
2 2
3 + 1 =
2 2
3 + 1. 3 − 1
3 − 1 =
2 2( 3 − 1)
3 − 1
= 2( 3 − 1) = 6 − 2.
e) 𝑠𝑒𝑐 75° = 1
𝑐𝑜𝑠 75° =
2 2
3 − 1 =
2 2
3 − 1. 3 + 1
3 + 1 =
2 2( 3 + 1)
3 − 1
= 2( 3 + 1) = 6 + 2.
f) 𝑐𝑜𝑡 75° = 1
𝑡𝑎𝑛 75° =
1
2 + 3 =
1
2 + 3.
2 − 3
2 − 3 =
2 + 3
4 − 3 = 2 + 3.
48. Show that
𝑐𝑜𝑠 45° − 𝜃 . 𝑐𝑜𝑠 45° + 𝜃 − 𝑠𝑖𝑛 45° − 𝜃 . 𝑠𝑖𝑛 45° + 𝜃 = 0
Solution:
𝐿𝐻𝑆 = 𝑐𝑜𝑠 45° − 𝜃 . 𝑐𝑜𝑠 45° + 𝜃 − 𝑠𝑖𝑛 45° − 𝜃 . 𝑠𝑖𝑛 45° + 𝜃
= 𝑐𝑜𝑠 45° − 𝜃 + 45° + 𝜃
= 𝑐𝑜𝑠 45° − 𝜃 + 45° + 𝜃 = cos 90° = 0.
Page 19
49. Show that 𝑡𝑎𝑛 𝜋
4+ 𝜃 ∙ 𝑡𝑎𝑛
𝜋
4− 𝜃 = 1.
Solution: 𝐿𝐻𝑆 = 𝑡𝑎𝑛 𝜋
4 + 𝜃 . 𝑡𝑎𝑛
𝜋
4− 𝜃
= 𝑡𝑎𝑛
𝜋
4 + 𝑡𝑎𝑛𝜃
1 − 𝑡𝑎𝑛 𝜋
4.𝑡𝑎𝑛𝜃
𝑡𝑎𝑛
𝜋
4 − 𝑡𝑎𝑛𝜃
1 + 𝑡𝑎𝑛 𝜋
4.𝑡𝑎𝑛𝜃
= 1 + 𝑡𝑎𝑛𝜃
1 − 𝑡𝑎𝑛𝜃
1 − 𝑡𝑎𝑛𝜃
1 + 𝑡𝑎𝑛𝜃 = 1 = RHS.
50. Show that 𝑠𝑖𝑛 𝑥 + 𝑦 + 𝑠𝑖𝑛 𝑥 + 𝑦
𝑐𝑜𝑠 𝑥 + 𝑦 + 𝑐𝑜𝑠 𝑥 + 𝑦 = 𝑡𝑎𝑛 𝑥.
Solution: LHS =𝑠𝑖𝑛 𝑥 + 𝑦 + 𝑠𝑖𝑛 𝑥 − 𝑦
𝑐𝑜𝑠 𝑥 + 𝑦 + 𝑐𝑜𝑠 𝑥 − 𝑦
=𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑦 + 𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑦 + 𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑦 − 𝑐𝑜𝑠𝑥 𝑠𝑖𝑛𝑦
𝑐𝑜𝑠𝑥 𝑐𝑜𝑠𝑦 − 𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑦 + 𝑐𝑜𝑠𝑥 𝑐𝑜𝑠𝑦 + 𝑠𝑖𝑛𝑥 𝑠𝑖𝑛𝑦 =
2𝑠𝑖𝑛𝑥 𝑐𝑜𝑠𝑦
2𝑐𝑜𝑠𝑥 𝑐𝑜𝑠𝑦= tan 𝑥.
51. If 𝑡𝑎𝑛𝐴 = 1
2 , 𝑡𝑎𝑛𝐵 =
1
3 show that 𝐴 + 𝐵 =
𝜋
4
Solution: 𝑡𝑎𝑛 𝐴 + 𝐵 =𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 𝐵
1 − 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 𝐵 =
1
2 +
1
3
1 − 1
2 ×
1
3
= 5 6
1 − 1 6 =
5 6
5 6 = 1 = tan
π
4 ∴ 𝐴 + 𝐵 =
𝜋
4
52. Prove that 𝑡𝑎𝑛 85° − 𝑡𝑎𝑛 25° = 3(1 + 𝑡𝑎𝑛 85° ∙ 𝑡𝑎𝑛 25°
Solution:
𝑡𝑎𝑛 60° = 3 ⟹ 𝑡𝑎𝑛 85° − 25° = 3
⟹𝑡𝑎𝑛 85° − 𝑡𝑎𝑛 25°
1 + 𝑡𝑎𝑛 85°∙𝑡𝑎𝑛 25° = 3
⟹ 𝑡𝑎𝑛 85° − 𝑡𝑎𝑛 25° = 3(1 + 𝑡𝑎𝑛 85° ∙ 𝑡𝑎𝑛 25°)
53. Prove that a) 𝑡𝑎𝑛 69° + 𝑡𝑎𝑛 66°
1 − 𝑡𝑎𝑛 69°∙ 𝑡𝑎𝑛 66° = − 1 ; b)
𝑐𝑜𝑠 17° + 𝑠𝑖𝑛17°
𝑐𝑜𝑠 17°− 𝑠𝑖𝑛17°= 𝑡𝑎𝑛62°
Solution:
a) 𝐿𝐻𝑆 =𝑡𝑎𝑛 69° + 𝑡𝑎𝑛 66°
1 − 𝑡𝑎𝑛 69°∙ 𝑡𝑎𝑛 66° = 𝑡𝑎𝑛 69° + 66° = 𝑡𝑎𝑛 135°
= 𝑡𝑎𝑛 90° + 45° = − 𝑐𝑜𝑡 45° = − 1 = 𝑅𝐻𝑆 .
b) Dividing the numerator and denominator by 𝑐𝑜𝑠 17°, we get,
𝐿𝐻𝑆 =𝑐𝑜𝑠 17° + 𝑠𝑖𝑛 17°
𝑐𝑜𝑠 17° − 𝑠𝑖𝑛 17° =
1 + 𝑡𝑎 𝑛17°
1 − 𝑡𝑎𝑛 17°
= 𝑡𝑎𝑛 45° + 𝑡𝑎𝑛 17°
1 − 𝑡𝑎𝑛 45°∙ 𝑡𝑎𝑛 17° = 𝑡𝑎𝑛 45° + 17° = 𝑡𝑎𝑛 62° = 𝑅𝐻𝑆.
54. Prove that 𝑡𝑎𝑛 3𝐴 − 𝑡𝑎𝑛 2𝐴 − 𝑡𝑎𝑛 𝐴 = 𝑡𝑎𝑛 𝐴 ∙ 𝑡𝑎𝑛 2𝐴 ∙ 𝑡𝑎𝑛 3𝐴
Page 20
Solution: 𝑡𝑎𝑛 3𝐴 = 𝑡𝑎𝑛 𝐴 + 2𝐴 =𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 2𝐴
1 − 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 2𝐴
⟹ 𝑡𝑎𝑛 3𝐴 1 − 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 2𝐴 = 𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 2𝐴
⟹ 𝑡𝑎𝑛 3𝐴 − 𝑡𝑎𝑛 𝐴. 𝑡𝑎𝑛 2𝐴. 𝑡𝑎𝑛 3𝐴 = 𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 2𝐴
⟹ 𝑡𝑎𝑛 3𝐴 − 𝑡𝑎𝑛 2𝐴 − 𝑡𝑎𝑛 𝐴 = 𝑡𝑎𝑛 𝐴 ∙ 𝑡𝑎𝑛 2𝐴 ∙ 𝑡𝑎𝑛 3𝐴.
55. Prove that 𝑠𝑖𝑛 2𝐴 = 2𝑠𝑖𝑛 𝐴. 𝑐𝑜𝑠 𝐴.
Solution: We know that 𝑠𝑖𝑛 (𝑥 + 𝑦) = 𝑠𝑖𝑛 𝑥. 𝑐𝑜𝑠 𝑦 + 𝑐𝑜𝑠 𝑥. 𝑠𝑖𝑛 𝑦.
By taking 𝑥 = 𝑦 = 𝐴, we get,
𝑠𝑖𝑛 (𝐴 + 𝐴) = 𝑠𝑖𝑛 𝐴. 𝑐𝑜𝑠 𝐴 + 𝑐𝑜𝑠 𝐴. 𝑠𝑖𝑛 𝐴
⇒ 𝑠𝑖𝑛 2𝐴 = 2𝑠𝑖𝑛 𝐴. 𝑐𝑜𝑠 𝐴.
56. Prove that 𝑐𝑜𝑠 2𝐴 = 𝑐𝑜𝑠2𝐴 − 𝑠𝑖𝑛2 𝐴.
Solution: We know that 𝑐𝑜𝑠 (𝑥 + 𝑦) = 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 − 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦.
By taking 𝑥 = 𝑦 = 𝐴, we get,
𝑐𝑜𝑠 (𝐴 + 𝐴) = 𝑐𝑜𝑠 𝐴. 𝑐𝑜𝑠 𝐴 − 𝑠𝑖𝑛 𝐴. 𝑠𝑖𝑛 𝐴
⇒ 𝑐𝑜𝑠 2𝐴 = 𝑐𝑜𝑠2𝐴 − 𝑠𝑖𝑛2𝐴.
57. Prove that 𝑐𝑜𝑠 2𝐴 = 2. 𝑐𝑜𝑠2𝐴 − 1
Solution: We know that 𝑐𝑜𝑠 (𝑥 + 𝑦) = 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 − 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦.
By taking 𝑥 = 𝑦 = 𝐴, we get,
𝑐𝑜𝑠 (𝐴 + 𝐴) = 𝑐𝑜𝑠 𝐴. 𝑐𝑜𝑠 𝐴 − 𝑠𝑖𝑛 𝐴. 𝑠𝑖𝑛 𝐴
⇒ 𝑐𝑜𝑠 2𝐴 = 𝑐𝑜𝑠2𝐴 − 𝑠𝑖𝑛2𝐴
= 𝑐𝑜𝑠2𝐴 − (1 − 𝑐𝑜𝑠2𝐴)
= 𝑐𝑜𝑠2𝐴 − 1 + 𝑐𝑜𝑠2𝐴 = 𝑐𝑜𝑠2𝐴 − 1.
58. Prove that 𝑐𝑜𝑠 2𝐴 = 1 − 2. 𝑠𝑖𝑛2𝐴
Solution: We know that 𝑐𝑜𝑠 (𝑥 + 𝑦) = 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 − 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦.
By taking 𝑥 = 𝑦 = 𝐴, we get,
𝑐𝑜𝑠 (𝐴 + 𝐴) = 𝑐𝑜𝑠 𝐴. 𝑐𝑜𝑠 𝐴 − 𝑠𝑖𝑛 𝐴. 𝑠𝑖𝑛 𝐴
⇒ 𝑐𝑜𝑠 2𝐴 = 𝑐𝑜𝑠2𝐴 − 𝑠𝑖𝑛2𝐴
= (1 − 𝑠𝑖𝑛2𝐴) − 𝑠𝑖𝑛2𝐴 = 1 − 2𝑠𝑖𝑛2𝐴.
Page 21
59. Prove that 𝑡𝑎𝑛 2𝐴 = 2𝑡𝑎𝑛 𝐴
1 − 𝑡𝑎𝑛 2𝐴.
Solution: We know that 𝑡𝑎𝑛 (𝑥 + 𝑦) = 𝑡𝑎𝑛 𝑥 + 𝑡𝑎𝑛 𝑦
1 − 𝑡𝑎𝑛 𝑥 .𝑡𝑎𝑛 𝑦 .
By taking 𝑥 = 𝑦 = 𝐴, we get, 𝑡𝑎𝑛 (𝐴 + 𝐴) = 𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 𝐴
1 − 𝑡𝑎𝑛 𝐴.𝑡𝑎𝑛 𝐴
⇒ 𝑡𝑎𝑛 2𝐴 = 2𝑡𝑎𝑛 𝐴
1 − 𝑡𝑎𝑛 2𝐴.
60. Prove that 𝑠𝑖𝑛 2𝐴 = 2𝑡𝑎𝑛 𝐴
1 + 𝑡𝑎𝑛 2𝐴.
Solution: 2.𝑡𝑎𝑛 𝐴
1 + 𝑡𝑎𝑛 2 𝐴 =
2.𝑡𝑎𝑛 𝐴
𝑠𝑒𝑐 2 𝐴 = 2. 𝑡𝑎𝑛 𝐴. 𝑐𝑜𝑠2 𝐴 = 2
𝑠𝑖𝑛 𝐴
𝑐𝑜𝑠 𝐴𝑐𝑜𝑠2 𝐴
= 2. 𝑠𝑖𝑛 𝐴. 𝑐𝑜𝑠 𝐴 = 𝑠𝑖𝑛 2𝐴.
61. Prove that 𝑐𝑜𝑠 2𝐴 = 1 − 𝑡𝑎𝑛 2 𝐴
1 + 𝑡𝑎𝑛 2 𝐴.
Solution:
1 − 𝑡𝑎𝑛 2 𝐴
1 + 𝑡𝑎𝑛 2 𝐴 =
1 − 𝑡𝑎𝑛 2 𝐴
𝑠𝑒𝑐 2 𝐴 =
1
𝑠𝑒𝑐 2 𝐴 −
𝑡𝑎𝑛 2 𝐴
𝑠𝑒𝑐 2 𝐴 = 𝑐𝑜𝑠2 𝐴 − 𝑡𝑎𝑛2 𝐴. 𝑐𝑜𝑠2 𝐴
= 𝑐𝑜𝑠2 𝐴 − 𝑠𝑖𝑛2 𝐴
𝑐𝑜𝑠 2 𝐴𝑐𝑜𝑠2 𝐴 = 𝑐𝑜𝑠2 𝐴 − 𝑠𝑖𝑛2 𝐴 = 𝑐𝑜𝑠 2𝐴.
62. Prove that 𝑡𝑎𝑛2 𝐴 = 1 − 𝑐𝑜𝑠 2𝐴
1 + 𝑐𝑜𝑠 2𝐴.
Solution:
We know that 𝑐𝑜𝑠 2𝐴 = 1 − 2. 𝑠𝑖𝑛2𝐴. ∴ 2. 𝑠𝑖𝑛2𝐴 = 1 − 𝑐𝑜𝑠 2𝐴 ;
Also 𝑐𝑜𝑠 2𝐴 = 2. 𝑐𝑜𝑠2𝐴 − 1 ∴ 2. 𝑐𝑜𝑠2𝐴 = 1 + 𝑐𝑜𝑠 2𝐴.
∴ 1 − 𝑐𝑜𝑠 2𝐴
1 + 𝑐𝑜𝑠 2𝐴 =
2.𝑠𝑖𝑛 2𝐴
2.𝑐𝑜𝑠 2𝐴 = 𝑡𝑎𝑛2𝐴.
63. Find the values of a. 𝑠𝑖𝑛 221
2° ; b. 𝑐𝑜𝑠 22
1
2° ; c. 𝑡𝑎𝑛 22
1
2°.
Solution:
a. We know that 2. 𝑠𝑖𝑛2 𝜃
2 = 1 − 𝑐𝑜𝑠 𝜃.
By taking 𝜃 = 45° we get
2. 𝑠𝑖𝑛2 221
2° = 1 − 𝑐𝑜𝑠 45° = 1 −
1
2 =
2 − 1
2.
∴ 𝑠𝑖𝑛2 221
2° =
2 − 1
2 2 =
2( 2 − 1)
2 2. 2 =
2 − 2
4
∴ 𝑠𝑖𝑛 221
2° = 2 − 2
4 =
2 − 2
2
Page 22
b. We know that 2. 𝑐𝑜𝑠2 𝜃
2 = 1 + 𝑐𝑜𝑠 𝜃.
By taking 𝜃 = 45° we get
2. 𝑐𝑜𝑠2 221
2° = 1 + 𝑐𝑜𝑠 45° = 1 +
1
2 =
2 + 1
2.
∴ 𝑐𝑜𝑠2 221
2° =
2 + 1
2 2 =
2( 2 + 1)
2 2. 2 =
2 + 2
4.
∴ 𝑐𝑜𝑠 221
2° =
2 + 2
4 =
2 + 2
2.
c. We know that, 𝑡𝑎𝑛2 𝜃
2 =
1 − 𝑐𝑜𝑠 𝜃
1 + 𝑐𝑜𝑠 𝜃 .
By taking 𝜃 = 45° we get,
𝑡𝑎𝑛2 221
2° =
1 − 𝑐𝑜𝑠 45°
1 + 𝑐𝑜𝑠 45° =
1 − 1 2
1 + 1 2 =
2 − 1
2 + 1
= 2 − 1
2 + 1. 2 − 1
2 − 1 =
2 − 1 2
2 − 1 = 2 − 1
2.
∴ 𝑡𝑎𝑛 221
2° = 2 − 1.
64. If 𝑠𝑖𝑛 𝐴 =3
8 and 𝐴 is acute angle, find 𝑠𝑖𝑛 2𝐴.
Solution: Given 𝑠𝑖𝑛 𝐴 =3
8.
Since 𝐴 is acute,
𝑐𝑜𝑠 𝐴 = 1 − 𝑠𝑖𝑛2𝐴 = 1 − 9
64 =
64 − 9
64 =
55
8.
∴ 𝑠𝑖𝑛 2𝐴 = 2 𝑠𝑖𝑛 𝐴. 𝑐𝑜𝑠 𝐴 = 2 ∙3
8∙ 55
8 =
3 55
32.
65. Prove that 𝑐𝑜𝑠4𝐴 − 𝑠𝑖𝑛4𝐴 = cos 2𝐴.
Solution: 𝐿𝐻𝑆 = 𝑐𝑜𝑠2𝐴 2 − 𝑠𝑖𝑛2𝐴 2
= 𝑐𝑜𝑠2𝐴 + 𝑠𝑖𝑛2𝐴 𝑐𝑜𝑠2𝐴 − 𝑠𝑖𝑛2𝐴
= 1 ∙ cos 2𝐴 = cos 2𝐴 = 𝑅𝐻𝑆.
66. Prove that 2 + 2 + 2 𝑐𝑜𝑠 4𝜃 = 2. 𝑐𝑜𝑠 𝜃
Solution: 𝐿𝐻𝑆 = 2 + 2(1 + 𝑐𝑜𝑠 4𝜃) = 2 + 2.2𝑐𝑜𝑠2 2𝜃
= 2 + 2 𝑐𝑜𝑠 2𝜃 = 2(1 + 𝑐𝑜𝑠 2𝜃)
= 2 ∙ 2𝑐𝑜𝑠2 2𝜃 = 2𝑐𝑜𝑠 𝜃 = 𝑅𝐻𝑆.
Page 23
67. Prove that 1 + 𝑡𝑎𝑛 2(45° − 𝜃)
1 − 𝑡𝑎𝑛 2(45° − 𝜃) = 𝑐𝑜𝑠𝑒𝑐 2𝜃.
Solution: 1 + 𝑡𝑎𝑛 2(45° − 𝜃)
1 − 𝑡𝑎𝑛 2(45° − 𝜃) =
1 + 𝑡𝑎𝑛 2𝑡
1 − 𝑡𝑎𝑛 2𝑡 =
1
𝑐𝑜𝑠 2𝑡 where 𝑡 = 45° − 𝜃.
= 𝑠𝑒𝑐 2𝑡 = 𝑠𝑒𝑐 2 45° − 𝜃 = 𝑠𝑒𝑐 90° − 2𝜃 = 𝑐𝑜𝑠𝑒𝑐 2𝜃.
68. Prove that 8 𝑐𝑜𝑠3 𝜋
9 − 6 𝑐𝑜𝑠
𝜋
9 = 1.
Solution: 8 𝑐𝑜𝑠3 𝜋
9− 6 𝑐𝑜𝑠
𝜋
9 = 2 4 𝑐𝑜𝑠3
𝜋
9 − 3 𝑐𝑜𝑠
𝜋
9
= 2 𝑐𝑜𝑠 3 ∙𝜋
9 = 2 𝑐𝑜𝑠
𝜋
3 = 2 ∙
1
2 = 1.
69. Prove that 1 + 𝑠𝑖𝑛 𝜃 − 𝑐𝑜𝑠 𝜃
1 + 𝑠𝑖𝑛 𝜃 + 𝑐𝑜𝑠 𝜃 = 𝑡𝑎𝑛
𝜃
2
Solution: 1 + 𝑠𝑖𝑛 𝜃 − 𝑐𝑜𝑠 𝜃
1 + 𝑠𝑖𝑛 𝜃 + 𝑐𝑜𝑠 𝜃 =
1 − 𝑐𝑜𝑠 𝜃 + 𝑠𝑖𝑛 𝜃
1 + 𝑐𝑜𝑠 𝜃 + 𝑠𝑖𝑛 𝜃
=2𝑠𝑖𝑛 2
𝜃
2 + 2 𝑠𝑖𝑛 𝜃
2 ∙𝑐𝑜𝑠
𝜃
2
2𝑐𝑜𝑠 2 𝜃
2 + 2 𝑠𝑖𝑛 𝜃
2 ∙𝑐𝑜𝑠
𝜃
2
=2 𝑠𝑖𝑛 𝜃
2 𝑐𝑜𝑠
𝜃2 + 𝑠𝑖𝑛
𝜃2
2 𝑐𝑜𝑠 𝜃
2 𝑐𝑜𝑠
𝜃
2 + 𝑠𝑖𝑛
𝜃
2
= 𝑡𝑎𝑛 𝜃
2.
70. Prove that 𝑐𝑜𝑠 10° − 𝑐𝑜𝑠 50° = 𝑠𝑖𝑛 20°
Solution: 𝐿𝐻𝑆 = − 2 𝑠𝑖𝑛 10 ° + 50 °
2 𝑠𝑖𝑛
10° − 50°
2
= − 2 𝑠𝑖𝑛 30° 𝑠𝑖𝑛 −20° = − (− 2 ∙1
2𝑠𝑖𝑛 20°) = 𝑠𝑖𝑛20°.
71. Find the general solution of 𝑠𝑖𝑛 𝜃 =1
2
Solution: 𝑠𝑖𝑛 𝜃 =1
2= 𝑠𝑖𝑛
𝜋
6. Principal angle 𝛼 =
𝜋
6.
∴ the general solution is 𝑥 = 𝑛𝜋 + (−1)𝑛 ∙ 𝜋
6 ,𝑛 ∈ 𝑍.
72. Find the general solution of 𝑠𝑒𝑐 𝜃 = − 2.
Solution: 𝑠𝑒𝑐 𝜃 = − 2 ⟹ cos 𝜃 = −1
2.
Principal angle 𝛼 = 𝜋 −𝜋
4 =
3𝜋
4
∴ general solution is 𝜃 = 2𝑛𝜋 ± 3𝜋
4 , 𝑛 ∈ 𝑍.
73. Find the general solution of 𝑠𝑖𝑛 2𝑥 = 4 𝑐𝑜𝑠 𝑥.
Page 24
Solution: 𝑠𝑖𝑛 2𝑥 = 4 𝑐𝑜𝑠 𝑥 ⟹ 2𝑠𝑖𝑛 𝑥 . 𝑐𝑜𝑠 𝑥 = 4 𝑐𝑜𝑠 𝑥
⟹ 2𝑠𝑖𝑛 𝑥. 𝑐𝑜𝑠 𝑥 − 4𝑐𝑜𝑠 𝑥 = 0 ⟹ 2𝑐𝑜𝑠 𝑥(𝑠𝑖𝑛 𝑥 − 2) = 0
⟹ 𝑐𝑜𝑠 𝑥 = 0 𝑜𝑟 𝑠𝑖𝑛 𝑥 = 2.
Since 𝑠𝑖𝑛 𝑥 = 2 > 1, the equation 𝑠𝑖𝑛 𝑥 = 2 has no solution.
𝑐𝑜𝑠 𝑥 = 0 ⟹ 𝑥 = 2𝑛 + 1 𝜋
2 , 𝑛 ∈ Z.
This is the general solution of the given equation.
Three mark questions
1. The angles of a triangle are in the ratio 4: 5: 6. Find them in
radian and degree.
Solution: Let A, B and C be the angles of the triangle.
Now 𝐴: 𝐵: 𝐶 = 4: 5: 6 ⟹ 𝐴 = 4𝜃, 𝐵 = 5𝜃,𝐶 = 6𝜃
𝐴 + 𝐵 + 𝐶 = 𝜋 ⟹ 4𝜃 + 5𝜃 + 6𝜃 = 𝜋 ⟹ 15𝜃 = 𝜋 ∴ 𝜃 =𝜋
15
∴ 𝐴 = 4𝜃 = 4 ×𝜋
15=
4𝜋
15 , 𝐵 = 5𝜃 = 5 ×
𝜋
15=
𝜋
3, 𝐶 = 6𝜃 = 6 ×
𝜋
15=
2𝜋
3.
𝐴 + 𝐵 + 𝐶 = 180° ⟹ 15𝜃 = 180° ∴ 𝜃 =180°
15= 12°
∴ 𝐴 = 4𝜃 = 4 × 12° = 48°, 𝐵 = 5 × 12° = 60°, 𝐶 = 6𝜃 = 6 × 12° = 72°
2. Define the six trigonometric functions.
Solution: Consider a circle of radius 𝑟 centered at the origin
of the Cartesian coordinate system. Let
𝑃(𝑥, 𝑦) be a point on the circle. Join 𝑂𝑃.
Define angle 𝜃 having 𝑂𝑋 has the initial
direction and 𝑂𝑃 as the terminal direction.
Then for any position of the point 𝑃(𝑥, 𝑦)
on the circle,
i. sine of the angle 𝜃 is denoted by 𝑠𝑖𝑛 𝜃 and is defined by
𝑠𝑖𝑛 𝜃 = 𝑦
𝑟.
ii. cosine of the angle 𝜃 is denoted by 𝑐𝑜𝑠 𝜃 and is defined by
𝑐𝑜𝑠 𝜃 = 𝑥
𝑟.
iii. tangent of the angle 𝜃 is denoted by 𝑡𝑎𝑛 𝜃 and is defined by
Page 25
𝑡𝑎𝑛 𝜃 = 𝑦
𝑥.
iv. cotangent of the angle 𝜃 is denoted by 𝑐𝑜𝑡 𝜃 and is defined
by 𝑐𝑜𝑡 𝜃 = 𝑥
𝑦.
v. secant of the angle 𝜃 is denoted by 𝑠𝑒𝑐 𝜃 and is defined by
𝑠𝑒𝑐 𝜃 = 𝑟
𝑥.
vi. cosecant of the angle 𝜃 is denoted by 𝑐𝑜𝑠𝑒𝑐 𝜃 or 𝑐𝑠𝑐 𝜃 and
is defined by 𝑐𝑜𝑠𝑒𝑐 𝜃 = 𝑟
𝑦.
3. If sin 𝜃 =3
5 and 𝜃 is acute, find all the other trigonometric
ratios.
Solution: Given sin 𝜃 = 3
5=
𝐴𝐵
𝑂𝐴.
Therefore 𝐴𝐵 = 3 and 𝑂𝐴 = 5
Now 𝑂𝐵2 = 𝑂𝐴2 − 𝐴𝐵2 = 25 − 9 = 16.
Therefore 𝑂𝐵 = 4.
We have, cos𝜃 = 1 − 𝑠𝑖𝑛2𝜃 = 1 − 9
25 =
25 − 9
25 =
16
25=
4
5 ;
tan 𝜃 =sin 𝜃
cos 𝜃=
35
45
=3
4 ;
cot 𝜃 =cos 𝜃
sin 𝜃=
45
35
=4
3 ;
𝑠𝑒𝑐𝜃 =1
cos 𝜃=
14
5 =
5
4 ;
𝑐𝑜𝑠𝑒𝑐𝜃 =1
sin 𝜃=
13
5 =
5
3 .
4. Prove that 1 − 𝑠𝑖𝑛 𝐴 + 𝑐𝑜𝑠𝐴 2 = 2 1 − 𝑠𝑖𝑛 𝐴 1 + 𝑐𝑜𝑠 𝐴
Solution:
𝐿𝐻𝑆 = 1 − 𝑠𝑖𝑛 𝐴 + 𝑐𝑜𝑠𝐴 2
= 1 − 𝑠𝑖𝑛 𝐴 2 + 𝑐𝑜𝑠2 𝐴 + 2 1 − 𝑠𝑖𝑛 𝐴 ∙ 𝑐𝑜𝑠𝐴
= 1 − 𝑠𝑖𝑛 𝐴 2 + 1 − 𝑠𝑖𝑛2 𝐴 + 2 1 − 𝑠𝑖𝑛 𝐴 ∙ 𝑐𝑜𝑠𝐴
= 1 − 𝑠𝑖𝑛 𝐴 2 + 1 − 𝑠𝑖𝑛 𝐴 ∙ 1 + 𝑠𝑖𝑛 𝐴 + 2 1 − 𝑠𝑖𝑛 𝐴 ∙ 𝑐𝑜𝑠𝐴
= 1 − 𝑠𝑖𝑛 𝐴 1 − 𝑠𝑖𝑛 𝐴 + 1 + 𝑠𝑖𝑛 𝐴 + 2 𝑐𝑜𝑠 𝐴
= 1 − 𝑠𝑖𝑛 𝐴 2 + 2 𝑐𝑜𝑠 𝐴 = 2 1 − 𝑠𝑖𝑛 𝐴 1 + 𝑐𝑜𝑠 𝐴 = 𝑅𝐻𝑆.
Page 26
5. Find the trigonometric ratios of the quadrantal angles
a) 0° ; b) 90° 𝜋
2 ; c) 180° (𝜋) ; d) 270°
3𝜋
2 and e) 360° (2𝜋).
Solution: Consider the points, 𝐴(1, 0), 𝐵(0, 1), 𝐶(− 1, 0) and
𝐷(0, −1) corresponding to the quadrantal angles 𝜋
2 (90°),
𝜋 (180°),3𝜋
2 (270°) and 2𝜋 (360°),
lying on a circle of unit radius
centered at the origin of a Cartesian
co-ordinate system. Let 𝑃(𝑎, 𝑏) be
the point on the circle such that
angle 𝑃𝑂𝑋 = 𝑥.
Then 𝑠𝑖𝑛 𝑥 = 𝑏 and 𝑐𝑜𝑠 𝑥 = 𝑎.
a)When 𝑃(𝑎, 𝑏) coincides with the point 𝐴(1, 0), the angle 𝑥
becomes 0°.
∴ 𝑐𝑜𝑠 0 = 1 and 𝑠𝑖𝑛 0 = 0.
∴ 𝑡𝑎𝑛 0 = 0
1 = 0, 𝑐𝑜𝑡 0 =
1
0 = ∞, 𝑐𝑜𝑠𝑒𝑐 0 =
1
0 = ∞, 𝑠𝑒𝑐 0 =
1
1 = 1.
b) When 𝑃(𝑎, 𝑏) coincides with the point 𝐵(0, 1), the angle 𝑥
becomes 90°.
∴ 𝑐𝑜𝑠 90° = 0 and 𝑠𝑖𝑛 90° = 1.
∴ 𝑡𝑎𝑛 90° = 1
0 = ∞, 𝑐𝑜𝑡 90° =
0
1 = 0,
𝑐𝑜𝑠𝑒𝑐 90° = 1
1 = 1, 𝑠𝑒𝑐 90° =
1
0 = ∞.
c) When 𝑃(𝑎, 𝑏) coincides with the point 𝐶(− 1, 0), the angle 𝑥
becomes 180°.
∴ 𝑐𝑜𝑠 180° = − 1 and 𝑠𝑖𝑛 180° = 0.
∴ 𝑡𝑎𝑛 180° = 0
− 1 = 0, 𝑐𝑜𝑡 180° =
− 1
0 = ∞,
𝑐𝑜𝑠𝑒𝑐 180° = 1
0 = ∞, 𝑠𝑒𝑐 180° =
1
− 1 = − 1.
d) When 𝑃(𝑎, 𝑏) coincides with the point 𝐷(0, −1), the angle 𝑥
becomes 270°.
∴ 𝑐𝑜𝑠 270° = 0 and 𝑠𝑖𝑛 270° = − 1.
Page 27
∴ 𝑡𝑎𝑛 270° = − 1
0 = ∞, 𝑐𝑜𝑡 270° =
0
− 1 = 0,
𝑐𝑜𝑠𝑒𝑐 270° = 1
− 1 = − 1, 𝑠𝑒𝑐 270° =
1
0 = ∞.
e) When 𝑃(𝑎, 𝑏) coincides with the point 𝐴(1, 0), after one
complete rotation, the angle 𝑥 becomes 360°.
∴ 𝑐𝑜𝑠 360° = 1 and 𝑠𝑖𝑛 360° = 0.
∴ 𝑡𝑎𝑛 360° = 0
1 = 0, 𝑐𝑜𝑡 360° =
1
0 = ∞,
𝑐𝑜𝑠𝑒𝑐 360° = 1
0 = ∞, 𝑠𝑒𝑐 360° =
1
1 = 1.
6. Find the trigonometric ratios of a) 45°; b) 30° and 60°.
Solution:
a) The trigonometric ratios of 45°.
Consider a right angled triangle 𝐴𝐵𝐶, right angled
at 𝐵. Now angle 𝐴𝐶𝐵 = 45°.
Let 𝐴𝐵 = 𝑥. Then 𝐵𝐶 = 𝑥.
Now 𝐴𝐶2 = 𝐴𝐵2 + 𝐵𝐶2 = 𝑥2 + 𝑥2 = 2𝑥2.
∴ 𝐴𝐶 = 𝑥 2. Now,
𝑠𝑖𝑛 45° = 𝐴𝐵
𝐴𝐶=
𝑥
𝑥 2=
1
2 ;
𝑐𝑜𝑠 45° = 𝐵𝐶
𝐴𝐶=
𝑥
𝑥 2=
1
2 ;
𝑡𝑎𝑛 45° = 𝐴𝐵
𝐵𝐶=
𝑥
𝑥= 1.
Similarly, 𝑐𝑜𝑡 45° = 1, 𝑐𝑜𝑠𝑒𝑐 45° = 2, 𝑠𝑒𝑐 45° = 2.
b) The trigonometric ratios of 30° and 60°.
Consider an equilateral triangle 𝐴𝐵𝐶. Draw 𝐴𝐷
perpendicular to 𝐵𝐶. Then 𝐷 is the midpoint of
𝐵𝐶. Let 𝐵𝐷 = 𝑥. Then 𝐴𝐵 = 2𝑥.
Now 𝐴𝐷2 = 𝐴𝐵2 − 𝐵𝐷2 = (2𝑥)2 − 𝑥2 = 4𝑥2 −
𝑥2 = 3𝑥2. ∴ 𝐴𝐷 = 𝑥 3.
Also, ∠𝐴𝐵𝐶 = 60° and ∠𝐵𝐴𝐷 = 30°.
𝑠𝑖𝑛 60° = 𝐴𝐷
𝐴𝐵=
𝑥 3
2𝑥=
3
2 ;
𝑐𝑜𝑠 60° = 𝐵𝐷
𝐴𝐵=
𝑥
2𝑥=
1
2 ;
Page 28
𝑡𝑎𝑛 60° = 𝐴𝐷
𝐵𝐷=
𝑥 3
𝑥= 3.
Similarly, 𝑐𝑜𝑡 60° = 1
3, 𝑐𝑜𝑠𝑒𝑐 60° =
2
3, 𝑠𝑒𝑐 60° = 2.
𝑠𝑖𝑛 30° = 𝐵𝐷
𝐴𝐵 =
𝑥
2𝑥 =
1
2 ;
𝑐𝑜𝑠 30° = 𝐴𝐷
𝐴𝐵 =
𝑥 3
2𝑥 =
3
2 ;
𝑡𝑎𝑛 30° = 𝐵𝐷
𝐴𝐷 =
𝑥
𝑥 3 =
1
3.
Similarly, 𝑐𝑜𝑡 30° = 3, 𝑐𝑜𝑠𝑒𝑐 30° = 2, 𝑠𝑒𝑐 30° = 2
3.
7. Prove that 𝑡𝑎𝑛 260° − 2 𝑡𝑎𝑛 245°
3 𝑠𝑖𝑛 245° ∙ 𝑠𝑖𝑛90° + 𝑐𝑜𝑠 260° ∙ 𝑐𝑜𝑠 2 0° =
4
7
Solution: 𝐿𝐻𝑆 = 3
2 − 2 1 2
3 1
2
2∙1 +
1
2
2∙ 1 2
=3 − 23
2 +
1
4
= 17
4
= 4
7 = 𝑅𝐻𝑆.
8. Find 𝑥, 𝑥 sin 30° ∙ 𝑐𝑜𝑠245° =𝑐𝑜𝑡 230°∙ 𝑠𝑒𝑐60°∙𝑡𝑎𝑛 45°
𝑐𝑜𝑠𝑒𝑐 245° ∙ 𝑐𝑜𝑠𝑒𝑐 30°
Solution: 𝑥 ∙1
2∙
1
2
2=
3 2∙ 2 ∙ 1
2 2∙ 2
⟹ 𝑥 ∙1
4 =
6
4. ∴ 𝑥 = 6.
9. Find 𝑥, 𝑥 𝑠𝑖𝑛230° + 𝑠𝑖𝑛245° + 𝑠𝑖𝑛260° − 𝑐𝑜𝑠230° = 0
Solution: 𝑥 1
2
2+
1
2
2+
3
2
2
− 3
2
2
= 0
⟹ 𝑥 1
4 +
1
2 +
3
4 −
3
4= 0 ⟹ 𝑥
1 + 2 + 3
4 =
3
4 ∴ 𝑥 =
3
6 =
1
2 .
10. Find the domain of 𝑓(𝑥) = 𝑠𝑖𝑛 𝑥 and 𝑓(𝑥) = 𝑐𝑜𝑠𝑒𝑐 𝑥.
Solution: Recall the definition, 𝑠𝑖𝑛 𝜃 = 𝑦
𝑟.
This is defined for any 𝑦 ∈ 𝑅.
∴ domain of the function 𝑦 = 𝑠𝑖𝑛 𝑥 is the set of all reals, 𝑅.
We know that 𝑠𝑖𝑛 0 = 0, 𝑠𝑖𝑛 𝜋 = 0 and 𝑠𝑖𝑛 2𝜋 = 0.
By using 𝑠𝑖𝑛 (2𝜋𝑛 + 𝜃) = 𝑠𝑖𝑛 𝜃 we get
𝑠𝑖𝑛 (2𝜋𝑛 + 𝜋) = 0, ∀ 𝑛 ∈ 𝑍.
This gives 𝑠𝑖𝑛 𝑛𝜋 = 0, ∀ 𝑛 ∈ 𝑍.
Thus 𝑠𝑖𝑛 𝑥 = 0 when 𝑥 = 𝑛𝜋, 𝑛 ∈ 𝑍.
Since 𝑐𝑜𝑠𝑒𝑐 𝑥 = 1
𝑠𝑖𝑛 𝑥, the function 𝑦 = 𝑐𝑜𝑠𝑒𝑐 𝑥 is defined only
Page 29
when 𝑠𝑖𝑛 𝑥 ≠ 0. Thus the domain of the function,
𝑦 = 𝑐𝑜𝑠𝑒𝑐 𝑥, is {𝑥 ∶ 𝑥 ≠ 𝑛𝜋, 𝑛 ∈ 𝑍 }.
11. Find the domain of 𝑓(𝑥) = 𝑐𝑜𝑠 𝑥 and 𝑓(𝑥) = 𝑠𝑒𝑐 𝑥.
Solution: Recall the definition, 𝑐𝑜𝑠 𝜃 = 𝑥
𝑟.
This is defined for any 𝑥 ∈ 𝑅.
∴ domain of the function 𝑦 = 𝑐𝑜𝑠 𝑥 is the set of all reals, 𝑅.
We know that 𝑐𝑜𝑠 𝜋
2 = 0, 𝑐𝑜𝑠
3𝜋
2 = 0.
By using 𝑐𝑜𝑠 (2𝜋𝑛 + 𝜃) = 𝑐𝑜𝑠 𝜃 we get
𝑐𝑜𝑠 (2𝜋𝑛 +𝜋
2) = 0, ∀ 𝑛 ∈ 𝑍.
This gives 𝑐𝑜𝑠 2𝑛 + 1 𝜋
2 = 0, ∀ 𝑛 ∈ 𝑍.
Thus 𝑐𝑜𝑠 𝑥 = 0 when 𝑥 = 2𝑛 + 1 𝜋
2, 𝑛 ∈ 𝑍.
Since 𝑠𝑒𝑐 𝑥 = 1
𝑐𝑜𝑠 𝑥, the function 𝑦 = 𝑠𝑒𝑐 𝑥 is defined only when
𝑐𝑜𝑠 𝑥 ≠ 0. Thus the domain of the function,
𝑦 = 𝑠𝑒𝑐 𝑥, is {𝑥 ∶ 𝑥 ∈ 𝑅, 𝑥 ≠ 2𝑛 + 1 𝜋
2, 𝑛 ∈ 𝑍 }.
12. Find the domain of 𝑓(𝑥) = 𝑡𝑎𝑛 𝑥.
Solution: Recall the definition, 𝑡𝑎𝑛 𝜃 = 𝑦
𝑥. This is defined for
any 𝑥, 𝑦 ∈ 𝑅, 𝑥 ≠ 0. We know that, 𝑡𝑎𝑛 𝜋
2 = ∞, 𝑡𝑎𝑛
3𝜋
2 = ∞.
By using 𝑡𝑎𝑛 (2𝜋𝑛 + 𝜃) = 𝑡𝑎𝑛 𝜃 we get 𝑡𝑎𝑛 (2𝜋𝑛 +𝜋
2) = ∞,
∀ 𝑛 ∈ 𝑍. This gives 𝑡𝑎𝑛 2𝑛 + 1 𝜋
2 = ∞, ∀ 𝑛 ∈ 𝑍.
Thus 𝑡𝑎𝑛 𝑥 = ∞ when 𝑥 = 2𝑛 + 1 𝜋
2, 𝑛 ∈ 𝑍. Thus the domain
of the function, 𝑦 = 𝑡𝑎𝑛 𝑥, is {𝑥 ∶ 𝑥 ∈ 𝑅, 𝑥 ≠ 2𝑛 + 1 𝜋
2, 𝑛 ∈ 𝑍 }.
13. Find the domain of 𝑓(𝑥) = 𝑐𝑜𝑡 𝑥.
Solution: Recall the definition, 𝑐𝑜𝑡 𝜃 = 𝑥
𝑦. This is defined for
any 𝑥, 𝑦 ∈ 𝑅, 𝑦 ≠ 0. We know that 𝑐𝑜𝑡 𝜋 = ∞, 𝑐𝑜𝑡 3𝜋 = ∞.
By using 𝑐𝑜𝑡 (2𝜋𝑛 + 𝜃) = 𝑐𝑜𝑡 𝜃 we get 𝑐𝑜𝑡 (2𝜋𝑛 + 𝜋) = ∞, ∀ 𝑛 ∈
𝑍. This gives 𝑐𝑜𝑡 𝑛𝜋 = ∞, ∀ 𝑛 ∈ 𝑍. Thus 𝑐𝑜𝑡 𝑥 = ∞ when
Page 30
𝑥 = 𝑛𝜋, 𝑛 ∈ 𝑍. Thus the domain of the function, 𝑦 = 𝑐𝑜𝑡 𝑥, is
{𝑥 ∶ 𝑥 ∈ 𝑅, 𝑥 ≠ 𝑛𝜋, 𝑛 ∈ 𝑍 }.
14. Prove that 𝑠𝑖𝑛 (𝑥 + 𝑦) = 𝑠𝑖𝑛 𝑥. 𝑐𝑜𝑠 𝑦 + 𝑐𝑜𝑠 𝑥. 𝑠𝑖𝑛 𝑦. Hence prove
that 𝑠𝑖𝑛 (𝑥 − 𝑦) = 𝑠𝑖𝑛 𝑥. 𝑐𝑜𝑠 𝑦 − 𝑐𝑜𝑠 𝑥. 𝑠𝑖𝑛 𝑦.
Solution: 𝑠𝑖𝑛 (𝑥 + 𝑦) = 𝑐𝑜𝑠 𝜋
2 − (𝑥 + 𝑦)
= 𝑐𝑜𝑠 𝜋
2 − 𝑥 − 𝑦
= 𝑐𝑜𝑠 𝜋
2 − 𝑥 . 𝑐𝑜𝑠 𝑦 + 𝑠𝑖𝑛
𝜋
2 − 𝑥 . 𝑠𝑖𝑛 𝑦
= 𝑠𝑖𝑛 𝑥. 𝑐𝑜𝑠 𝑦 + 𝑐𝑜𝑠 𝑥. 𝑠𝑖𝑛 𝑦.
By replacing 𝑦 by − 𝑦 we get,
𝑠𝑖𝑛 (𝑥 − 𝑦) = 𝑠𝑖𝑛 𝑥. 𝑐𝑜𝑠 (− 𝑦) + 𝑐𝑜𝑠 𝑥. 𝑠𝑖𝑛 ( − 𝑦)
= 𝑠𝑖𝑛 𝑥. 𝑐𝑜𝑠 𝑦 + 𝑐𝑜𝑠 𝑥. (− 𝑠𝑖𝑛 𝑦)
= 𝑠𝑖𝑛 𝑥. 𝑐𝑜𝑠 𝑦 − 𝑐𝑜𝑠 𝑥. 𝑠𝑖𝑛 𝑦.
15. Draw the graph of 𝑦 = 𝑠𝑖𝑛 𝑥.
Solution: 𝑠𝑖𝑛 0 = 0 and 𝑠𝑖𝑛 𝜋 2 = 1.
∴ 𝑠𝑖𝑛 𝑥 increases from 0 to 1 as 𝑥 increases from 0 to 𝜋 2 .
𝑠𝑖𝑛 𝜋 2 = 1 and 𝑠𝑖𝑛 𝜋 = 0.
∴ 𝑠𝑖𝑛 𝑥 decreases from 1 to 0 as 𝑥 increases from 𝜋 2 to 𝜋.
𝑠𝑖𝑛 𝜋 = 0 and 𝑠𝑖𝑛 3𝜋 2 = − 1.
∴ 𝑠𝑖𝑛 𝑥 decreases from 0 to − 1 as 𝑥 increases from 𝜋 to 3𝜋 2 .
𝑠𝑖𝑛 3𝜋 2 = −1 and 𝑠𝑖𝑛 2𝜋 = 0.
∴ 𝑠𝑖𝑛 𝑥 increases from − 1 to 0 as 𝑥 increases from 3𝜋 2 to 2𝜋.
Since 𝑠𝑖𝑛 𝑥 is periodic with the period 2𝜋 we can make the
same observation in the
intervals
..., −4𝜋,− 2𝜋 , −2𝜋, 0 ,
0, 2𝜋 , 2𝜋, 4𝜋 , 4𝜋, 6𝜋 ,
...The graph is shown in the adjacent figure.
Page 31
16. Draw the graph of 𝑦 = 𝑐𝑜𝑠 𝑥.
Solution: 𝑐𝑜𝑠 0 = 1 and 𝑐𝑜𝑠 𝜋 2 = 0.
∴ 𝑐𝑜𝑠 𝑥 decreases from 1 to 0 as 𝑥 increases from 0 to 𝜋 2 .
𝑐𝑜𝑠 𝜋 2 = 0 and 𝑐𝑜𝑠 𝜋 = − 1.
∴ 𝑐𝑜𝑠 𝑥 decreases from 0 to − 1 as 𝑥 increases from 𝜋 2 to 𝜋.
𝑐𝑜𝑠 𝜋 = − 1 and 𝑐𝑜𝑠 3𝜋 2 = 0.
∴ 𝑐𝑜𝑠 𝑥 increases from − 1 to 0 as 𝑥 increases from 𝜋 to 3𝜋 2 .
𝑐𝑜𝑠 3𝜋 2 = 0 and 𝑐𝑜𝑠 2𝜋 = 1.
∴ 𝑐𝑜𝑠 𝑥 increases from 0 to 1 as 𝑥 increases from 3𝜋 2 to 2𝜋.
Since 𝑐𝑜𝑠 𝑥 is periodic with the period 2𝜋, we can make the
same observation in the
intervals
..., −4𝜋,− 2𝜋 , −2𝜋, 0 ,
0, 2𝜋 , 2𝜋, 4𝜋 , 4𝜋, 6𝜋 ,
... The graph is shown in the adjacent figure.
17. Draw the graph of 𝑦 = 𝑡𝑎𝑛 𝑥.
Solution: 𝑡𝑎𝑛 0 = 0 and 𝑡𝑎𝑛 𝜋 2 = ∞.
∴ 𝑡𝑎𝑛 𝑥 increases from 0 to ∞ as 𝑥 increases from 0 to 𝜋 2 .
𝑡𝑎𝑛 𝜋 2 = ∞ and 𝑡𝑎𝑛 𝜋 = 0.
Between 𝑥 = 𝜋 2 and 𝑥 = 𝜋, 𝑡𝑎𝑛 𝑥 is negative.
∴ 𝑡𝑎𝑛 𝑥 increases from − ∞ to 0 as 𝑥 increases from 𝜋 2 to
𝜋.
Since 𝑡𝑎𝑛 𝑥 is periodic with
the period 𝜋, we can make
the same observations in
the intervals, ... −2𝜋, − 𝜋 ,
−𝜋, 0 , 0, 𝜋 , 𝜋, 2𝜋 , ...
The graph is shown in the
adjacent figure.
Page 32
18. Draw the graph of 𝑦 = 𝑐𝑜𝑡 𝑥.
Solution: 𝑐𝑜𝑡 0 = ∞ and 𝑐𝑜𝑡 𝜋 2 = 0.
∴ 𝑐𝑜𝑡 𝑥 decreases from ∞ to 0 as 𝑥 increases from 0 to 𝜋 2 .
𝑐𝑜𝑡 𝜋 2 = 0 and 𝑐𝑜𝑡 𝜋 = ∞.
Between 𝑥 = 𝜋 2 and 𝑥 = 𝜋 𝑐𝑜𝑡 𝑥 is negative.
∴ 𝑐𝑜𝑡 𝑥 decreases from 0 to
− ∞ as 𝑥 increases from 𝜋 2
to 𝜋.
Since 𝑐𝑜𝑡 𝑥 is periodic with the
period 𝜋, we can make the
same observations in the
intervals, ... −2𝜋, − 𝜋 , −𝜋, 0 ,
0, 𝜋 , 𝜋, 2𝜋 , ... The graph is shown in the adjacent figure.
19. Draw the graph of 𝑦 = 𝑐𝑜𝑠𝑒𝑐 𝑥.
Solution: 𝑐𝑜𝑠𝑒𝑐 0 = ∞ and 𝑐𝑜𝑠𝑒𝑐 𝜋 2 = 1.
∴ 𝑐𝑜𝑠𝑒𝑐 𝑥 decreases from ∞ to 0 as 𝑥 increases from 0 to 𝜋 2 .
𝑐𝑜𝑠𝑒𝑐 𝜋 2 = 1 and 𝑐𝑜𝑠𝑒𝑐 𝜋 = ∞.
∴ 𝑐𝑜𝑠𝑒𝑐 𝑥 increases from 1 to ∞ as 𝑥 increases from 𝜋 2 to 𝜋.
𝑐𝑜𝑠𝑒𝑐 𝜋 = ∞ and 𝑐𝑜𝑠𝑒𝑐 3𝜋 2 = − 1.
Between 𝑥 = 𝜋 and 𝑥 = 3𝜋 2 , 𝑐𝑜𝑠𝑒𝑐 𝑥 is negative.
∴ 𝑐𝑜𝑠𝑒𝑐 𝑥 increases from
− ∞ to − 1 as 𝑥 increases
from 𝜋 to 3 𝜋 2 .
𝑐𝑜𝑠𝑒𝑐 3𝜋 2 = − 1 and
𝑐𝑜𝑠𝑒𝑐 2𝜋 = ∞.
Between 𝑥 = 3𝜋 2 and
𝑥 = 2𝜋, 𝑐𝑜𝑠𝑒𝑐 𝑥 is negative.
∴ 𝑐𝑜𝑠𝑒𝑐 𝑥 decreases from − 1 to − ∞ as 𝑥 increases from
3 𝜋 2 to 2𝜋.
Since 𝑐𝑜𝑠𝑒𝑐 𝑥 is periodic with the period 2𝜋, we can make the
same observations in the intervals, ..., −4𝜋, − 2𝜋 , − 2𝜋, 0 ,
0, 2𝜋 , 2𝜋, 4𝜋 , ... The graph is shown in the adjacent figure.
Page 33
20. Draw the graph of 𝑦 = 𝑠𝑒𝑐 𝑥.
Solution: 𝑠𝑒𝑐 0 = 1 and 𝑠𝑒𝑐 𝜋 2 = ∞.
∴ 𝑠𝑒𝑐 𝑥 increases from 1 to ∞ as 𝑥 increases from 0 to 𝜋 2 .
𝑠𝑒𝑐 𝜋 2 = ∞ and 𝑠𝑒𝑐 𝜋 = − 1.
Between 𝑥 = 𝜋 2 and 𝑥 = 𝜋, 𝑠𝑒𝑐 𝑥 is negative.
∴ 𝑠𝑒𝑐 𝑥 increases from − ∞ to − 1 as 𝑥 increases from 𝜋 2 to 𝜋.
𝑠𝑒𝑐 𝜋 = − 1 and 𝑠𝑒𝑐 3𝜋 2 = ∞.
Between 𝑥 = 𝜋 and 𝑥 = 3𝜋 2 , 𝑠𝑒𝑐 𝑥 is negative. ∴ 𝑠𝑒𝑐 𝑥
decreases from − 1 to − ∞. as 𝑥 increases from 𝜋 to 3 𝜋 2 .
𝑠𝑒𝑐 3𝜋 2 = ∞ and 𝑠𝑒𝑐 2𝜋 = 1.
Between 𝑥 = 3𝜋 2 and
𝑥 = 2𝜋, 𝑠𝑒𝑐 𝑥 is positive.
∴ 𝑠𝑒𝑐 𝑥 decreases from ∞
to 1 as 𝑥 increases from
3 𝜋 2 to 2𝜋.
Since 𝑠𝑒𝑐 𝑥 is periodic with
the period 2𝜋, we can make
the same observations in the
intervals, ..., −4𝜋, − 2𝜋 , − 2𝜋, 0 , 0, 2𝜋 , 2𝜋, 4𝜋 , ... The
graph is shown in the adjacent figure.
21. If 𝑐𝑜𝑠 𝜃 =4
5 and 𝜃 is acute, find the value of
3𝑐𝑜𝑠 𝜃 + 2𝑐𝑜𝑠𝑒𝑐 𝜃
4 𝑠𝑖𝑛 𝜃 − 𝑐𝑜𝑡 𝜃 .
Solution: Given cos 𝜃 =4
5. Let 𝑎𝑑𝑗 = 4 and ℎ𝑦𝑝 = 5. Then
𝑜𝑝𝑝 = 3. Then 𝑐𝑜𝑠𝑒𝑐 𝜃 = ℎ𝑦𝑝
𝑜𝑝𝑝 =
5
3 and 𝑐𝑜𝑡 𝜃 =
𝑎𝑑𝑗
𝑜𝑝𝑝 =
4
3.
Therefore 3𝑐𝑜𝑠 𝜃 + 2𝑐𝑜𝑠𝑒𝑐 𝜃
4 𝑠𝑖𝑛 𝜃 − 𝑐𝑜𝑡 𝜃 =
3 4
5 + 2
5
3
4 3
5 −
4
3
= 12
5 +
10
312
5 −
4
3
= 86
1516
15
=86
16 =
43
8.
Page 34
22. If 𝑠𝑒𝑐 𝜃 =13
5 and
3𝜋
2< 𝜃 < 2𝜋, find the value of
2𝑠𝑖𝑛 𝜃 − 3𝑐𝑜𝑠 𝜃
4 𝑠𝑖𝑛 𝜃 + 9 𝑐𝑜𝑠 𝜃 .
Solution: Given 𝑠𝑒𝑐 𝜃 =13
5. Let ℎ𝑦𝑝 = 13 and 𝑎𝑑𝑗 = 5.
Then 𝑜𝑝𝑝 = 12. In the fourth quadrant 𝑠𝑖𝑛 𝜃 = − 𝑜𝑝𝑝
ℎ𝑦𝑝 = −
12
13 ;
Thus 2𝑠𝑖𝑛 𝜃 − 3𝑐𝑜𝑠 𝜃
4 𝑠𝑖𝑛 𝜃 + 9 𝑐𝑜𝑠 𝜃 =
2sin 𝜃 − 3cos 𝜃
4 sin 𝜃 9 cos 𝜃=
2 −12
13 − 3
5
13
4 −12
13 + 9
5
13
=−24
13 −
15
13−48
13 +
45
13
=−39
−3= 13.
23. If 𝐴, 𝐵 are acute angles, 𝑠𝑖𝑛 𝐴 =3
5 ; 𝑐𝑜𝑠 𝐵 =
12
13, find 𝑐𝑜𝑠 𝐴 + 𝐵
Solution: We have 𝑐𝑜𝑠 𝐴 + 𝐵 = 𝑐𝑜𝑠𝐴 𝑐𝑜𝑠𝐵 − 𝑠𝑖𝑛𝐴 𝑠𝑖𝑛𝐵.
Given 𝑠𝑖𝑛 𝐴 =3
5 ; 𝑐𝑜𝑠 𝐵 =
12
13.
∴ 𝑐𝑜𝑠 𝐴 = 1 − 𝑠𝑖𝑛2𝐴 = 1 −9
25 =
25 − 9
25 =
4
5 and
𝑠𝑖𝑛𝐵 = 1 − 𝑐𝑜𝑠2𝐵 = 1 −144
169 =
5
13.
∴ 𝑐𝑜𝑠 𝐴 + 𝐵 = 4
5×
12
13 −
3
5×
5
13 =
48 − 15
65 =
33
65.
24. If 𝐴 + 𝐵 = 45° then prove that 1 + 𝑡𝑎𝑛 𝐴 1 + 𝑡𝑎𝑛 𝐵 = 2 and
hence deduce that 𝑡𝑎𝑛 221
2
°= 2 − 1.
Solution: Given 𝐴 + 𝐵 = 45° ⟹ 𝑡𝑎𝑛 𝐴 + 𝐵 = 𝑡𝑎𝑛 45°
⟹ 𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 𝐵
1 − 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 𝐵 = 1 ⟹ 𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 𝐵 = 1 − 𝑡𝑎𝑛 𝐴. 𝑡𝑎𝑛 𝐵
⟹ 𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 𝐵 + 𝑡𝑎𝑛 𝐴. 𝑡𝑎𝑛 𝐵 = 1 (by adding 1 to both sides)
⟹ 1 + 𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 𝐵 + 𝑡𝑎𝑛 𝐴. 𝑡𝑎𝑛 𝐵 = 2
⟹ 1 + 𝑡𝑎𝑛 𝐴 + 𝑡𝑎𝑛 𝐵. 1 + 𝑡𝑎𝑛 𝐴 = 2
⟹ 1 + 𝑡𝑎𝑛 𝐴 . 1 + 𝑡𝑎𝑛 𝐵 = 2.
By taking 𝐴 = 𝐵 = 221
2
° we get,
1 + 𝑡𝑎𝑛 221
2
°
2
= 2 ⟹ 1 + 𝑡𝑎𝑛 221
2
°= ± 2
⟹ 𝑡𝑎𝑛 221
2
°= ± 2 − 1.
Since 221
2
° is acute, 𝑡𝑎𝑛 221
2
° is positive. Therefore
𝑡𝑎𝑛 221
2
°= 2 − 1.
Page 35
25. Prove that 𝑠𝑖𝑛 3𝑥 = 3𝑠𝑖𝑛 𝑥 − 4𝑠𝑖𝑛3 𝑥
Solution: 𝑠𝑖𝑛 3𝑥 = 𝑠𝑖𝑛 (2𝑥 + 𝑥) = 𝑠𝑖𝑛 2𝑥. 𝑐𝑜𝑠 𝑥 + 𝑐𝑜𝑠 2𝑥. 𝑠𝑖𝑛 𝑥
= (2𝑠𝑖𝑛 𝑥. 𝑐𝑜𝑠 𝑥)𝑐𝑜𝑠 𝑥 + (1 − 2𝑠𝑖𝑛2 𝑥)𝑠𝑖𝑛 𝑥
= 2. 𝑠𝑖𝑛 𝑥. 𝑐𝑜𝑠2 𝑥 + 𝑠𝑖𝑛 𝑥 − 2𝑠𝑖𝑛3 𝑥
= 2. 𝑠𝑖𝑛 𝑥(1 − 𝑠𝑖𝑛2 𝑥) + 𝑠𝑖𝑛 𝑥 − 2𝑠𝑖𝑛3 𝑥 = 3𝑠𝑖𝑛 𝑥 − 4𝑠𝑖𝑛3 𝑥.
26. Prove that 𝑐𝑜𝑠 3𝑥 = 4𝑐𝑜𝑠3 𝑥 − 3𝑐𝑜𝑠 𝑥
Solution: 𝑐𝑜𝑠 3𝑥 = 𝑐𝑜𝑠 (2𝑥 + 𝑥) = 𝑐𝑜𝑠 2𝑥. 𝑐𝑜𝑠 𝑥 − 𝑠𝑖𝑛 2𝑥. 𝑠𝑖𝑛 𝑥
= (2𝑐𝑜𝑠2𝑥 − 1)𝑐𝑜𝑠 𝑥 − (2𝑠𝑖𝑛 𝑥. 𝑐𝑜𝑠 𝑥)𝑠𝑖𝑛 𝑥
= 2𝑐𝑜𝑠3 𝑥 − 𝑐𝑜𝑠 𝑥 − 2𝑠𝑖𝑛2 𝑥. 𝑐𝑜𝑠 𝑥
= 2𝑐𝑜𝑠3 𝑥 − 𝑐𝑜𝑠 𝑥 − 2(1 − 𝑐𝑜𝑠2 𝑥)𝑐𝑜𝑠 𝑥
= 2𝑐𝑜𝑠3 𝑥 − 𝑐𝑜𝑠 𝑥 − 2𝑐𝑜𝑠 𝑥 + 2𝑐𝑜𝑠3 𝑥 = 4𝑐𝑜𝑠3 𝑥 − 3𝑐𝑜𝑠 𝑥.
27. Prove that 𝑡𝑎𝑛 3𝑥 = 3𝑡𝑎𝑛 𝑥 − 𝑡𝑎𝑛 3 𝑥
1 − 3𝑡𝑎𝑛 2 𝑥
Solution: 𝑡𝑎𝑛 3𝑥 = 𝑡𝑎𝑛 (2𝑥 + 𝑥) =𝑡𝑎𝑛 2𝑥 + 𝑡𝑎𝑛 𝑥
1 − 𝑡𝑎𝑛 2𝑥 .𝑡𝑎𝑛 𝑥
= 2𝑡𝑎𝑛 𝑥
1 − 𝑡𝑎𝑛 2 𝑥 + 𝑡𝑎𝑛 𝑥
1 − 2𝑡𝑎𝑛 𝑥
1 − 𝑡𝑎𝑛 2 𝑥 .𝑡𝑎𝑛 𝑥
= 2𝑡𝑎𝑛 𝑥 + 𝑡𝑎𝑛 𝑥 (1 − 𝑡𝑎𝑛 2 𝑥)
1 − 𝑡𝑎𝑛 2 𝑥 − 2𝑡𝑎𝑛 2 𝑥
= 2𝑡𝑎𝑛 𝑥 + 𝑡𝑎𝑛 𝑥 − 𝑡𝑎𝑛 3 𝑥)
1 − 𝑡𝑎𝑛 2 𝑥 − 2𝑡𝑎𝑛 2 𝑥 =
3𝑡𝑎𝑛 𝑥 − 𝑡𝑎𝑛 3 𝑥
1 − 3𝑡𝑎𝑛 2 𝑥.
28. If 𝑡𝑎𝑛 𝐴 =1 − 𝑐𝑜𝑠 𝐵
𝑠𝑖𝑛 𝐵, prove that 𝑡𝑎𝑛 2𝐴 = 𝑡𝑎𝑛 𝐵, where 𝐴 and 𝐵
are acute angles.
Solution: 𝑅𝐻𝑆 =1 − 𝑐𝑜𝑠 𝐵
𝑠𝑖𝑛 𝐵 =
2𝑠𝑖𝑛 2 𝐵
2
2 𝑠𝑖𝑛 𝐵2 ∙𝑐𝑜𝑠
𝐵
2
=𝑠𝑖𝑛 𝐵
2
𝑐𝑜𝑠 𝐵
2
= 𝑡𝑎𝑛 𝐵2 .
∴ 𝑡𝑎𝑛 𝐴 = 𝑡𝑎𝑛 𝐵2 .
Since 𝐴 and 𝐵 are acute angles we get, 𝐴 = 𝐵
2 . ∴ 2𝐴 = 𝐵
∴ 𝑡𝑎𝑛 2𝐴 = 𝑡𝑎𝑛 𝐵.
Page 36
29. Show that 4 𝑠𝑖𝑛 𝐴. 𝑠𝑖𝑛 60° + 𝐴 . 𝑠𝑖𝑛 60° − 𝐴 = 𝑠𝑖𝑛 3𝐴.
Solution: 𝐿𝐻𝑆 = 4 𝑠𝑖𝑛 𝐴. {𝑠𝑖𝑛 60° + 𝐴 . 𝑠𝑖𝑛 60° − 𝐴 }
= 4 𝑠𝑖𝑛 𝐴 {𝑠𝑖𝑛2 60° − 𝑠𝑖𝑛2 𝐴} = 4 𝑠𝑖𝑛 𝐴 3
2
2
− 𝑠𝑖𝑛2 𝐴
= 4 𝑠𝑖𝑛 𝐴 3
4− 𝑠𝑖𝑛2𝐴 = 3𝑠𝑖𝑛 𝐴 − 4𝑠𝑖𝑛3𝐴 = 𝑠𝑖𝑛3𝐴 = 𝑅𝐻𝑆.
30. Show that 4 𝑐𝑜𝑠 𝐴. 𝑐𝑜𝑠 60° + 𝐴 . 𝑐𝑜𝑠 60° − 𝐴 = 𝑐𝑜𝑠 3𝐴.
Solution: 𝐿𝐻𝑆 = 4 𝑐𝑜𝑠 𝐴. 𝑐𝑜𝑠 60° + 𝐴 . 𝑐𝑜𝑠 60° − 𝐴
= 4 𝑐𝑜𝑠 𝐴 {𝑐𝑜𝑠2 60° − 𝑠𝑖𝑛2 𝐴}
= 4 𝑐𝑜𝑠 𝐴 1
4− 1 − 𝑐𝑜𝑠2𝐴 = 4 𝑐𝑜𝑠 𝐴 −
3
4+ 𝑐𝑜𝑠2𝐴
= − 3𝑐𝑜𝑠 𝐴 + 4𝑐𝑜𝑠3𝐴 = 4𝑐𝑜𝑠3𝐴 − 3𝑐𝑜𝑠 𝐴 = 𝑐𝑜𝑠3𝐴 = 𝑅𝐻𝑆.
31. Prove that 2 𝑠𝑖𝑛 𝐴 − 𝑠𝑖𝑛 2𝐴
2 𝑠𝑖𝑛 𝐴 + 𝑠𝑖𝑛 2𝐴 = 𝑡𝑎𝑛2
𝐴
2.
Solution: 2 𝑠𝑖𝑛 𝐴 − 𝑠𝑖𝑛 2𝐴
2 𝑠𝑖𝑛 𝐴 + 𝑠𝑖𝑛 2𝐴 =
2 𝑠𝑖𝑛 𝐴 − 2𝑠𝑖𝑛 𝐴.𝑐𝑜𝑠 𝐴
2 𝑠𝑖𝑛 𝐴 + 2𝑠𝑖𝑛 𝐴.𝑐𝑜𝑠 𝐴
= 2 𝑠𝑖𝑛 𝐴(1 − 𝑐𝑜𝑠 𝐴)
2 𝑠𝑖𝑛 𝐴(1 + 𝑐𝑜𝑠 𝐴) =
2𝑠𝑖𝑛 2 𝐴
2
2𝑐𝑜𝑠 2 𝐴
2
= 𝑡𝑎𝑛2 𝐴
2.
32. Prove that 𝑠𝑖𝑛 4𝑥 𝑐𝑜𝑠 2𝑥
1 + 𝑐𝑜𝑠 4𝑥 1 + 𝑐𝑜𝑠 2𝑥 = 𝑡𝑎𝑛 𝑥
Solution: 𝑠𝑖𝑛 4𝑥 𝑐𝑜𝑠 2𝑥
1 + 𝑐𝑜𝑠 4𝑥 1 + 𝑐𝑜𝑠 2𝑥 =
2𝑠𝑖𝑛 2𝑥 .𝑐𝑜𝑠 2𝑥 .𝑐𝑜𝑠 2𝑥
2𝑐𝑜𝑠 2 2𝑥 .2𝑐𝑜𝑠 2 𝑥 =
𝑠𝑖𝑛 2𝑥
2𝑐𝑜𝑠 2 𝑥
= 2𝑠𝑖𝑛 𝑥 .𝑐𝑜𝑠 𝑥
2𝑐𝑜𝑠 𝑥 .𝑐𝑜𝑠 𝑥 = 𝑡𝑎𝑛 𝑥.
33. If 𝑠𝑖𝑛 𝐴 = 3
5 and 𝐴 is acute angle, find the values of 𝑠𝑖𝑛 2𝐴
and 𝑐𝑜𝑠 2𝐴.
Solution: 𝑠𝑖𝑛 𝐴 = 3
5 ⇒ 𝑜𝑝𝑝 = 3 and ℎ𝑦𝑝 = 5. ∴ 𝑎𝑑𝑗 = 4.
∴ 𝑐𝑜𝑠 𝐴 = 𝑎𝑑𝑗
ℎ𝑦𝑝=
4
5.
∴ 𝑠𝑖𝑛 2𝐴 = 2𝑠𝑖𝑛 𝐴. 𝑐𝑜𝑠 𝐴 = 23
5.
4
5 =
24
25 ;
𝑐𝑜𝑠 2𝐴 = 1 − 2𝑠𝑖𝑛2𝐴 = 1 − 2 3
5
2 = 1 − 2
9
25 = 1 −
18
25 =
7
25.
Page 37
34. If 𝑡𝑎𝑛 𝐴 = 1
3, 𝑡𝑎𝑛 𝐵 =
1
7 prove that 2𝐴 + 𝐵 =
𝜋
4.
Solution: 𝑡𝑎𝑛 2𝐴 = 2𝑡𝑎𝑛 𝐴
1 − 𝑡𝑎𝑛 2 𝐴 =
2 1 3
1 − 1 3 2 =
2 3
1 − 1 9 =
2 3
8 9 =
3
4.
∴ 𝑡𝑎𝑛 (2𝐴 + 𝐵) = 𝑡𝑎𝑛 2𝐴 + 𝑡𝑎𝑛 𝐵
1 − 𝑡𝑎𝑛 2𝐴.𝑡𝑎𝑛 𝐵 =
3
4 +
1
7
1 − 3
4
1
7
= 21 + 4
2828 − 3
28
= 25
25 = 1.
Since 𝑡𝑎𝑛 𝐴 and 𝑡𝑎𝑛 𝐵 are positive and since 𝑡𝑎𝑛 (2𝐴 + 𝐵) is
positive, we get, 2𝐴 + 𝐵 = 𝜋
4.
35. Show that 𝑡𝑎𝑛 𝐴 =1 − 𝑐𝑜𝑠 2𝐴
𝑠𝑖𝑛 2𝐴 and hence deduce the value of
𝑡𝑎𝑛15°.
Solution: 1 − 𝑐𝑜𝑠 2𝐴
𝑠𝑖𝑛 2𝐴 =
2𝑠𝑖𝑛 2 𝐴
2𝑠𝑖𝑛 𝐴.𝑐𝑜𝑠 𝐴 = 𝑡𝑎𝑛 𝐴. ∴ 𝑡𝑎𝑛 𝐴 =
1 − 𝑐𝑜𝑠 2𝐴
𝑠𝑖𝑛 2𝐴.
By taking 𝐴 = 15° we get,
𝑡𝑎𝑛 15° =1 − 𝑐𝑜𝑠 2(15°)
𝑠𝑖𝑛 2(15°) =
1 − 𝑐𝑜𝑠 30°
𝑠𝑖𝑛 30° =
1 − 3 2
1 2 =
2 − 3
2
1 2 = 2 − 3.
36. Prove that 𝑐𝑜𝑠 20° + 𝑐𝑜𝑠 100° + 𝑐𝑜𝑠 140° = 0.
Solution: 𝐿𝐻𝑆 = 𝑐𝑜𝑠 20° + 𝑐𝑜𝑠 100° + 𝑐𝑜𝑠 140°
= 𝑐𝑜𝑠 20° + 2 𝑐𝑜𝑠 100 ° + 140 °
2 . 𝑐𝑜𝑠
100° − 140°
2
= 𝑐𝑜𝑠 20° + 2 𝑐𝑜𝑠 120° . 𝑐𝑜𝑠 −20°
= 𝑐𝑜𝑠 20° + 2 −1
2 . 𝑐𝑜𝑠 20° = 𝑐𝑜𝑠 20° − 𝑐𝑜𝑠 20° = 0.
37. Prove that 𝑠𝑖𝑛 50° − 𝑠𝑖𝑛 70° + 𝑠𝑖𝑛 10° = 0
Solution: 𝐿𝐻𝑆 = 𝑠𝑖𝑛 50° − 𝑠𝑖𝑛 70° + 𝑠𝑖𝑛 10°
= 2 𝑐𝑜𝑠 50 ° + 70°
2 . 𝑠𝑖𝑛
50° − 70°
2 + 𝑠𝑖𝑛10°
= 2 𝑐𝑜𝑠 60° ∙ 𝑠𝑖𝑛 −10° + 𝑠𝑖𝑛10°
= − 2 ×1
2𝑠𝑖𝑛 10° + 𝑠𝑖𝑛 10° = − 𝑠𝑖𝑛 10° + 𝑠𝑖𝑛 10° = 0.
38. Prove that 𝑠𝑖𝑛 20° 𝑠𝑖𝑛 40° 𝑠𝑖𝑛 60° 𝑠𝑖𝑛 80° =3
16
Solution: 𝐿𝐻𝑆 = 𝑠𝑖𝑛 60° . 𝑠𝑖𝑛 20° 𝑠𝑖𝑛(60° − 20°) 𝑠𝑖𝑛(60° + 20°)
= 3
2𝑠𝑖𝑛 20° 𝑠𝑖𝑛 60° − 20° 𝑠𝑖𝑛 60° + 20°
Page 38
= 3
2.
1
4. sin (3 × 20°) =
3
2.
1
4. sin 60°
= 3
8 𝑠𝑖𝑛 60° =
3
8×
3
2=
3
16.
39. Show that 𝑠𝑖𝑛 5𝐴 + 𝑠𝑖𝑛 2𝐴 − 𝑠𝑖𝑛 𝐴
𝑐𝑜𝑠 5𝐴 + 𝑐𝑜𝑠 2𝐴 + 𝑐𝑜𝑠 𝐴 = 𝑡𝑎𝑛 2𝐴
Solution: 𝑠𝑖𝑛 5𝐴 + 𝑠𝑖𝑛 2𝐴 − 𝑠𝑖𝑛 𝐴
𝑐𝑜𝑠 5𝐴 + 𝑐𝑜𝑠 2𝐴 + 𝑐𝑜𝑠 𝐴=
𝑠𝑖𝑛 2𝐴 + 𝑠𝑖𝑛 5𝐴 − 𝑠𝑖𝑛 𝐴
𝑐𝑜𝑠 2𝐴 + 𝑐𝑜𝑠 5𝐴 + 𝑐𝑜𝑠 𝐴
=𝑠𝑖𝑛 2𝐴 + 2 𝑐𝑜𝑠 5𝐴 + 𝐴
2 .𝑠𝑖𝑛
5𝐴 − 𝐴
2
𝑐𝑜𝑠 2𝐴 + 2 𝑐𝑜𝑠 5𝐴 + 𝐴2
.𝑐𝑜𝑠 5𝐴 − 𝐴
2
=𝑠𝑖𝑛 2𝐴 + 2 𝑐𝑜𝑠 3𝐴 . 𝑠𝑖𝑛 2𝐴
𝑐𝑜𝑠 2𝐴 + 2 𝑐𝑜𝑠 3𝐴 . 𝑐𝑜𝑠 2𝐴
=𝑠𝑖𝑛 2𝐴[1 + 2 𝑐𝑜𝑠 3𝐴]
𝑐𝑜𝑠 2𝐴[1 + 2 𝑐𝑜𝑠 3𝐴] =
𝑠𝑖𝑛 2𝐴
𝑐𝑜𝑠 2𝐴= 𝑡𝑎𝑛 2𝐴.
40. Find the general solution of 𝑠𝑖𝑛 𝜃 = 𝑘, −1 ≤ 𝑘 ≤ 1.
Solution: Let 𝛼 be an angle such that
𝑠𝑖𝑛 𝛼 = 𝑘, −𝜋
2 ≤ 𝛼 ≤
𝜋
2.
Then 𝑠𝑖𝑛 𝜃 = 𝑠𝑖𝑛 𝛼.
Now 𝑠𝑖𝑛 𝜃 = 𝑠𝑖𝑛 𝛼 ⇒ 𝑠𝑖𝑛 𝜃 – 𝑠𝑖𝑛 𝛼 = 0
⇒ 2𝑐𝑜𝑠 𝜃 + 𝛼
2. 𝑠𝑖𝑛
𝜃 − 𝛼
2 = 0
⇒ 𝑐𝑜𝑠 𝜃 + 𝛼
2 = 0 or 𝑠𝑖𝑛
𝜃 − 𝛼
2 = 0.
Now 𝑐𝑜𝑠 𝜃 + 𝛼
2 = 0 ⇒
𝜃 + 𝛼
2 = (2𝑛 + 1)
𝜋
2, 𝑛 ∈ 𝑍.
⇒ 𝜃 + 𝛼 = (2𝑛 + 1)𝜋
⇒ 𝜃 = (2𝑛 + 1)𝜋 − 𝛼 …(1)
𝑠𝑖𝑛 𝜃 − 𝛼
2 = 0 ⇒
𝜃 − 𝛼
2 = 𝑛𝜋, 𝑛 ∈ 𝑍.
⇒ 𝜃 − 𝛼 = 2𝑛𝜋 ⇒ 𝜃 = 2𝑛𝜋 + 𝛼 …(2).
∴ combining (1) and (2),we get, 𝜃 = 𝑛𝜋 + − 1 𝑛𝛼, 𝑛 ∈ 𝑍.
This is the general solution of 𝑠𝑖𝑛 𝜃 = 𝑘, − 1 ≤ 𝑘 ≤ 1.
41. Find the general solution of 𝑐𝑜𝑠 𝜃 = 𝑘, − 1 ≤ 𝑘 ≤ 1.
Solution: Let 𝛼 be an angle such that 𝑐𝑜𝑠 𝛼 = 𝑘, 0 ≤ 𝛼 ≤ 𝜋
Then 𝑐𝑜𝑠 𝜃 = 𝑐𝑜𝑠 𝛼.
Now 𝑐𝑜𝑠 𝜃 = 𝑐𝑜𝑠 𝛼 ⇒ 𝑐𝑜𝑠 𝜃 – 𝑐𝑜𝑠 𝛼 = 0
Page 39
⇒ − 2𝑠𝑖𝑛 𝜃 + 𝛼
2. 𝑠𝑖𝑛
𝜃 − 𝛼
2 = 0
⇒ 𝑠𝑖𝑛𝜃 + 𝛼
2 = 0 or 𝑠𝑖𝑛
𝜃 − 𝛼
2 = 0.
Now 𝑠𝑖𝑛 𝜃 + 𝛼
2 = 0 ⇒
𝜃 + 𝛼
2 = 𝑛𝜋, 𝑛 ∈ 𝑍
⇒ 𝜃 + 𝛼 = 2𝑛𝜋 ⇒ 𝜃 = 2𝑛𝜋 − 𝛼 …(1)
𝑠𝑖𝑛 𝜃 − 𝛼
2 = 0 ⇒
𝜃 − 𝛼
2 = 𝑛𝜋, 𝑛 ∈ 𝑍
⇒ 𝜃 − 𝛼 = 2𝑛𝜋 ⇒ 𝜃 = 2𝑛𝜋 + 𝛼 ...(2).
∴ combining (1) and (2), we get, 𝜃 = 2𝑛𝜋 ± 𝛼 , 𝑛 ∈ 𝑍.
This is the general solution of 𝑐𝑜𝑠 𝜃 = 𝑘, − 1 ≤ 𝑘 ≤ 1.
42. Find the general solution of 𝑡𝑎𝑛 𝜃 = 𝑘, − ∞ < 𝑘 < ∞.
Solution: Let 𝛼 be an angle such that
𝑡𝑎𝑛 𝛼 = 𝑘 , − 𝜋
2 < 𝛼 <
𝜋
2.
Then 𝑡𝑎𝑛 𝜃 = 𝑡𝑎𝑛 𝛼.
Now 𝑡𝑎𝑛 𝜃 = 𝑡𝑎𝑛 𝛼 ⇒ 𝑠𝑖𝑛𝜃
𝑐𝑜𝑠𝜃 =
𝑠𝑖𝑛𝛼
𝑐𝑜𝑠𝛼
⇒ 𝑠𝑖𝑛 𝜃. 𝑐𝑜𝑠 𝛼 = 𝑐𝑜𝑠 𝜃. 𝑠𝑖𝑛 𝛼
⇒ 𝑠𝑖𝑛 𝜃. 𝑐𝑜𝑠 𝛼 − 𝑐𝑜𝑠 𝜃. 𝑠𝑖𝑛 𝛼 = 0
⇒ 𝑠𝑖𝑛( 𝜃 – 𝛼 ) = 0 ⇒ 𝜃 – 𝛼 = 𝑛𝜋, 𝑛 ∈ 𝑍.
⇒ 𝜃 = 𝑛𝜋 + 𝛼 , 𝑛 ∈ 𝑍.
This is the general solution of 𝑡𝑎𝑛 𝜃 = 𝑘, − ∞ < 𝑘 < ∞.
43. Find the general solution of 𝑡𝑎𝑛 2𝜃. 𝑡𝑎𝑛 3𝜃 = 1.
Solution: 𝑡𝑎𝑛 2𝜃. 𝑡𝑎𝑛 3𝜃 = 1 ⟹ 𝑠𝑖𝑛 2 𝜃
𝑐𝑜𝑠 2 𝜃.𝑠𝑖𝑛 3 𝜃
𝑐𝑜𝑠 3 𝜃= 1
⟹ 𝑠𝑖𝑛 2𝜃. 𝑠𝑖𝑛 3𝜃 = 𝑐𝑜𝑠 2𝜃. 𝑐𝑜𝑠 3𝜃
⟹ 𝑐𝑜𝑠 2𝜃. 𝑐𝑜𝑠 3𝜃 − 𝑠𝑖𝑛 2𝜃. 𝑠𝑖𝑛 3𝜃 = 0
⟹ 𝑐𝑜𝑠 (2𝜃 + 3𝜃) = 0 ⟹ 𝑐𝑜𝑠 5𝜃 = 0.
⟹ 5 𝜃 = 2𝑛 + 1 𝜋
2, 𝑛 ∈ 𝑍 ⟹ 𝜃 = 2𝑛 + 1
𝜋
10, 𝑛 ∈ 𝑍.
This is the general solution of the given equation.
Page 40
44. Find the general solution of
𝑡𝑎𝑛 𝜃 + 𝑡𝑎𝑛 2𝜃 + 3𝑡𝑎𝑛 𝜃. 𝑡𝑎𝑛 2𝜃 = 3.
Solution: 𝑡𝑎𝑛 𝜃 + 𝑡𝑎𝑛 2𝜃 + 3𝑡𝑎𝑛 𝜃. 𝑡𝑎𝑛 2𝜃 = 3
⟹ 𝑡𝑎𝑛 𝜃 + 𝑡𝑎𝑛 2𝜃 = 3(1 − 𝑡𝑎𝑛 𝜃. 𝑡𝑎𝑛 2𝜃)
⟹ 𝑡𝑎𝑛 𝜃 + 𝑡𝑎𝑛 2𝜃
(1 − 𝑡𝑎𝑛 𝜃∙𝑡𝑎𝑛 2𝜃) = 3 ⟹ 𝑡𝑎𝑛 3𝜃 = 3 = 𝑡𝑎𝑛
𝜋
3
⟹ 3𝜃 = 𝑛𝜋 + 𝜋
3 , 𝑛 ∈ 𝑍 ⟹ 𝜃 =
𝑛𝜋
3 +
𝜋
9 , 𝑛 ∈ 𝑍.
This is the general solution of the given equation.
45. Find the general solution of 𝑠𝑖𝑛 2𝑥 + 𝑠𝑖𝑛 4𝑥 + 𝑠𝑖𝑛 6𝑥 = 0.
Solution: 𝑠𝑖𝑛 2𝑥 + 𝑠𝑖𝑛 4𝑥 + 𝑠𝑖𝑛 6𝑥 = 0
⟹ 𝑠𝑖𝑛 4𝑥 + 𝑠𝑖𝑛 6𝑥 + 𝑠𝑖𝑛 2𝑥 = 0
⟹ 𝑠𝑖𝑛 4𝑥 + 2𝑠𝑖𝑛 6𝑥 + 2𝑥
2 ∙ 𝑐𝑜𝑠
6𝑥 − 2𝑥
2 = 0
⟹ 𝑠𝑖𝑛 4𝑥 + 2𝑠𝑖𝑛 4𝑥. 𝑐𝑜𝑠 2𝑥 = 0 ⟹ 𝑠𝑖𝑛 4𝑥(1 + 2𝑐𝑜𝑠 2𝑥) = 0
⟹ 𝑠𝑖𝑛 4𝑥 = 0 or 1 + 2𝑐𝑜𝑠 2𝑥 = 0.
𝑠𝑖𝑛 4𝑥 = 0 ⟹ 4𝑥 = 𝑛𝜋 ∴ 𝑥 = 𝑛𝜋
4, 𝑛 ∈ 𝑍.
1 + 2𝑐𝑜𝑠 2𝑥 = 0 ⟹ 𝑐𝑜𝑠 2𝑥 = −1
2 = 𝑐𝑜𝑠
2𝜋
3.
∴ 2𝑥 = 2𝑛𝜋 ± 2𝜋
3, 𝑛 ∈ 𝑍. ∴ 𝑥 = 𝑛𝜋 ±
𝜋
3, 𝑛 ∈ 𝑍.
This is the general solution of the given equation.
46. Find the general solution of 4𝑠𝑖𝑛2 𝑥 − 8𝑐𝑜𝑠 𝑥 + 1 = 0.
Solution: 4𝑠𝑖𝑛2 𝑥 − 8𝑐𝑜𝑠 𝑥 + 1 = 0
⟹ 4(1 − 𝑐𝑜𝑠2 𝑥) − 8𝑐𝑜𝑠 𝑥 + 1 = 0
⟹ −4𝑐𝑜𝑠2 𝑥 − 8𝑐𝑜𝑠 𝑥 + 5 = 0
⟹ 4𝑐𝑜𝑠2 𝑥 + 8𝑐𝑜𝑠 𝑥 − 5 = 0
⟹ 4𝑐𝑜𝑠2 𝑥 + 10𝑐𝑜𝑠 𝑥 − 2𝑐𝑜𝑠 𝑥 − 5 = 0
⟹ 2𝑐𝑜𝑠 𝑥 2𝑐𝑜𝑠 𝑥 + 5 − 1 2𝑐𝑜𝑠 𝑥 + 5 = 0
⟹ 2𝑐𝑜𝑠 𝑥 − 1 2𝑐𝑜𝑠 𝑥 + 5 = 0
⟹ 2𝑐𝑜𝑠 𝑥 − 1 = 0 or 2𝑐𝑜𝑠 𝑥 + 5 = 0
⟹ 𝑐𝑜𝑠 𝑥 =1
2 or 𝑐𝑜𝑠 𝑥 = −
5
2
Now 𝑐𝑜𝑠𝑥 = − 5
2< −1. This has no solution.
Page 41
𝑐𝑜𝑠 𝑥 =1
2 ⟹ 𝑥 = 2𝑛𝜋 ±
𝜋
3 , 𝑛 ∈ 𝑍.
This is the general solution of the given equation.
47. Find the general solution of 2𝑠𝑖𝑛2 𝑥 + 3𝑐𝑜𝑠 𝑥 + 1 = 0.
Solution: 2𝑠𝑖𝑛2 𝑥 + 3𝑐𝑜𝑠 𝑥 + 1 = 0
⟹ 2(1 − 𝑐𝑜𝑠2 𝑥) + 3𝑐𝑜𝑠 𝑥 + 1 = 0
⟹ 2𝑐𝑜𝑠2 𝑥 − 2 3𝑐𝑜𝑠 𝑥 + 3𝑐𝑜𝑠 𝑥 − 3 = 0
⟹ 2𝑐𝑜𝑠 𝑥 𝑐𝑜𝑠 𝑥 − 3 + 3(𝑐𝑜𝑠 𝑥 − 3) = 0
⟹ (2𝑐𝑜𝑠 𝑥 + 3) 𝑐𝑜𝑠 𝑥 − 3 = 0
⟹ 𝑐𝑜𝑠 𝑥 = − 3
2 or 𝑐𝑜𝑠 𝑥 = 3.
𝑐𝑜𝑠𝑥 = 3 > 1. This has no solution.
Now 𝑐𝑜𝑠𝑥 = − 3
2 = cos 𝜋 −
𝜋
6 = 𝑐𝑜𝑠
5𝜋
6. ∴ 𝑥 = 2𝑛𝜋 ±
5𝜋
6, 𝑛 ∈ 𝑍.
This is the general solution of the given equation.
48. Find the general solution of 2𝑐𝑜𝑠2 𝑥 + 3𝑠𝑖𝑛 𝑥 = 0.
Solution: 2𝑐𝑜𝑠2 𝑥 + 3𝑠𝑖𝑛 𝑥 = 0
⟹ 2 1 − 𝑠𝑖𝑛2 𝑥 + 3𝑠𝑖𝑛 𝑥 = 0 ⟹ 2 − 2𝑠𝑖𝑛2 𝑥 + 3𝑠𝑖𝑛 𝑥 = 0
⟹ 2𝑠𝑖𝑛2 𝑥 − 3 𝑠𝑖𝑛 𝑥 − 2 = 0 ⟹ 2𝑠𝑖𝑛 𝑥 + 1 𝑠𝑖𝑛 𝑥 − 2 = 0
⟹ 2𝑠𝑖𝑛 𝑥 + 1 = 0 or 𝑠𝑖𝑛 𝑥 − 2 = 0
⟹ 𝑠𝑖𝑛 𝑥 = − 1
2 or 𝑠𝑖𝑛 𝑥 = 2
𝑠𝑖𝑛𝑥 = 2 > 1 has no solution.
𝑠𝑖𝑛 𝑥 = − 1
2 = −𝑠𝑖𝑛
𝜋
6= 𝑠𝑖𝑛 −
𝜋
6 .
∴ 𝑥 = 𝑛𝜋 + (−1)𝑛 −𝜋
6 , 𝑛 ∈ 𝑍.
This is the general solution of the given equation.
49. Find the general solution of 2𝑐𝑜𝑠2 𝜃 + 𝑠𝑖𝑛 𝜃 = 𝑠𝑖𝑛 2𝜃 + 𝑐𝑜𝑠 𝜃.
Solution: 2𝑐𝑜𝑠2 𝜃 + 𝑠𝑖𝑛 𝜃 = 𝑠𝑖𝑛 2𝜃 + 𝑐𝑜𝑠 𝜃
⟹ 2𝑐𝑜𝑠2 𝜃 − 𝑐𝑜𝑠 𝜃 = 2𝑠𝑖𝑛 𝜃𝑐𝑜𝑠 𝜃 − 𝑠𝑖𝑛 𝜃
⟹ 𝑐𝑜𝑠 𝜃 2𝑐𝑜𝑠 𝜃 − 1 − 𝑠𝑖𝑛 𝜃 2𝑐𝑜𝑠 𝜃 − 1 = 0
⟹ 2𝑐𝑜𝑠 𝜃 − 1 (𝑐𝑜𝑠 𝜃 − 𝑠𝑖𝑛 𝜃) = 0
⟹ 2𝑐𝑜𝑠 𝜃 − 1 = 0 or 𝑐𝑜𝑠 𝜃 − 𝑠𝑖𝑛 𝜃 = 0.
Page 42
Now 2𝑐𝑜𝑠 𝜃 − 1 = 0 ⟹ 𝑐𝑜𝑠 𝜃 = 1
2 = 𝑐𝑜𝑠
𝜋
3 ∴ 𝜃 = 2𝑛𝜋 ±
𝜋
3 , 𝑛 ∈ 𝑍.
𝑐𝑜𝑠 𝜃 − 𝑠𝑖𝑛 𝜃 = 0 ⟹ 𝑠𝑖𝑛 𝜃 = 𝑐𝑜𝑠 𝜃.
Dividing by 𝑐𝑜𝑠 𝜃 we get, 𝑡𝑎𝑛 𝜃 = 1 = 𝑡𝑎𝑛𝜋
4. ∴ 𝜃 = 𝑛𝜋 +
𝜋
4,
𝑛 ∈ 𝑍. ∴ the general solution is 𝜃 = 2𝑛𝜋 ± 𝜋
3, 𝜃 = 𝑛𝜋 +
𝜋
4, 𝑛 ∈ 𝑍.
50. Find the general solution of 𝑐𝑜𝑠 𝑥 + 𝑐𝑜𝑠 2𝑥 + 𝑐𝑜𝑠 3𝑥 = 0.
Solution: 𝑐𝑜𝑠 𝑥 + 𝑐𝑜𝑠 2𝑥 + 𝑐𝑜𝑠 3𝑥 = 0
⟹ 2 𝑐𝑜𝑠 𝑥 + 3𝑥
2 . 𝑐𝑜𝑠
𝑥 − 3𝑥
2 + 𝑐𝑜𝑠 2𝑥 = 0 ⟹ 𝑐𝑜𝑠 2𝑥 2𝑐𝑜𝑠 𝑥 + 1 = 0
⟹ 𝑐𝑜𝑠 2𝑥 = 0 or 2𝑐𝑜𝑠 𝑥 + 1 = 0.
Now 𝑐𝑜𝑠 2𝑥 = 0 ⟹ 2𝑥 = (2𝑛 + 1)𝜋
2 ∴ 𝑥 = 2𝑛 + 1
𝜋
4 ,𝑛 ∈ 𝑍.
2𝑐𝑜𝑠 𝑥 + 1 = 0 ⟹ 𝑐𝑜𝑠 𝑥 = −1
2= 𝑐𝑜𝑠 𝜋 −
𝜋
3 = 𝑐𝑜𝑠
2𝜋
3
∴ 𝑥 = 2𝑛𝜋 ±2𝜋
3 , 𝑛 ∈ 𝑍.
∴ the general solution is 𝑥 = 2𝑛 + 1 𝜋
4, 𝑥 = 2𝑛𝜋 ±
2𝜋
3 𝑛 ∈ 𝑍.
51. Prove that 2𝑐𝑜𝑠 𝜋
13. 𝑐𝑜𝑠
9𝜋
13 + 𝑐𝑜𝑠
3𝜋
13 + 𝑐𝑜𝑠
5𝜋
13 = 0.
Solution: 2𝑐𝑜𝑠 𝜋
13. 𝑐𝑜𝑠
9𝜋
13 + 𝑐𝑜𝑠
3𝜋
13 + 𝑐𝑜𝑠
5𝜋
13
= 2𝑐𝑜𝑠 𝜋
13. 𝑐𝑜𝑠
9𝜋
13 + 2𝑐𝑜𝑠
3𝜋
13 +
5𝜋
13
2. 𝑐𝑜𝑠
5𝜋
13 −
3𝜋
13
13
= 2𝑐𝑜𝑠 𝜋
13. 𝑐𝑜𝑠
9𝜋
13 + 2𝑐𝑜𝑠
8𝜋
13
2. 𝑐𝑜𝑠
2𝜋
13
2
= 2𝑐𝑜𝑠 𝜋
13. 𝑐𝑜𝑠
9𝜋
13 + 2𝑐𝑜𝑠
4𝜋
13. 𝑐𝑜𝑠
𝜋
13
= 2𝑐𝑜𝑠 𝜋
13. 𝑐𝑜𝑠
13𝜋 − 4𝜋
13 + 2𝑐𝑜𝑠
4𝜋
13. 𝑐𝑜𝑠
𝜋
13
= 2𝑐𝑜𝑠 𝜋
13. 𝑐𝑜𝑠 𝜋 −
4𝜋
13 + 2𝑐𝑜𝑠
4𝜋
13. 𝑐𝑜𝑠
𝜋
13
= − 2𝑐𝑜𝑠 𝜋
13. 𝑐𝑜𝑠
4𝜋
13 + 2𝑐𝑜𝑠
4𝜋
13. 𝑐𝑜𝑠
𝜋
13 = 0.
52. Prove that 𝑐𝑜𝑠 6𝑥 = 32𝑐𝑜𝑠6 𝑥 − 48𝑐𝑜𝑠4 𝑥 + 18𝑐𝑜𝑠2 𝑥 − 1
Solution: 𝑐𝑜𝑠 6𝑥 = 2𝑐𝑜𝑠2 3𝑥 − 1
= 2 4𝑐𝑜𝑠3 𝑥 − 3𝑐𝑜𝑠 𝑥 2 − 1
= 2 4𝑐𝑜𝑠3 𝑥 2 + 3𝑐𝑜𝑠 𝑥 2 − 24𝑐𝑜𝑠3 𝑥. 𝑐𝑜𝑠 𝑥 − 1
Page 43
= 2 16𝑐𝑜𝑠6 𝑥 + 9𝑐𝑜𝑠2 𝑥 − 24𝑐𝑜𝑠4 𝑥 − 1
= 32𝑐𝑜𝑠6 𝑥 − 48𝑐𝑜𝑠4 𝑥 + 18𝑐𝑜𝑠2 𝑥 − 1
53. Prove that 𝑐𝑜𝑡 𝑥. 𝑐𝑜𝑡 2𝑥 − 𝑐𝑜𝑡 2𝑥. 𝑐𝑜𝑡 3𝑥 − 𝑐𝑜𝑡 3𝑥. 𝑐𝑜𝑡 𝑥 = 1.
Solution: We have 𝑥 + 2𝑥 = 3𝑥.
∴ 𝑐𝑜𝑡 3𝑥 = 𝑐𝑜𝑡 (𝑥 + 2𝑥) = 𝑐𝑜𝑡 𝑥 .𝑐𝑜𝑡 2𝑥 − 1
𝑐𝑜𝑡 2𝑥 + 𝑐𝑜𝑡 𝑥.
∴ 𝑐𝑜𝑡 3𝑥(𝑐𝑜𝑡 2𝑥 − 𝑐𝑜𝑡 𝑥) = 𝑐𝑜𝑡 𝑥. 𝑐𝑜𝑡 2𝑥 + 1
∴ 𝑐𝑜𝑡 3𝑥. 𝑐𝑜𝑡 2𝑥 + 𝑐𝑜𝑡 3𝑥. 𝑐𝑜𝑡 𝑥 = 𝑐𝑜𝑡 𝑥. 𝑐𝑜𝑡 2𝑥 − 1
Thus 𝑐𝑜𝑡 3𝑥. 𝑐𝑜𝑡 2𝑥 + 𝑐𝑜𝑡 3𝑥. 𝑐𝑜𝑡 𝑥 − 𝑐𝑜𝑡 𝑥. 𝑐𝑜𝑡 2𝑥 = − 1.
By changing the sign we get
𝑐𝑜𝑡 𝑥. 𝑐𝑜𝑡 2𝑥 − 𝑐𝑜𝑡 2𝑥. 𝑐𝑜𝑡 3𝑥 − 𝑐𝑜𝑡 3𝑥. 𝑐𝑜𝑡 𝑥 = 1.
Five mark questions
1. Discuss the signs of trigonometric functions in the four
quadrants.
Solution: Consider a circle of radius 𝑟,
centered at the origin of a Cartesian
coordinate system. Let 𝑃(𝑥, 𝑦) be a
point on the circle. Suppose that the
angle made by the radius 𝑂𝑃 with the
positive direction of the
x-axis is 𝜃. Then
𝑠𝑖𝑛 𝜃 = 𝑦
𝑟, 𝑐𝑜𝑠 𝜃 =
𝑥
𝑟, 𝑡𝑎𝑛 𝜃 =
𝑦
𝑥 .
a) If 0 < 𝜃 < 𝜋
2 then 𝑃(𝑥, 𝑦 ) is a point in the first quadrant.
Here 𝑥 > 0 and 𝑦 > 0. ∴ 𝑠𝑖𝑛 𝜃 > 0, 𝑐𝑜𝑠 𝜃 > 0, 𝑡𝑎𝑛 𝜃 > 0
∴ 𝑐𝑜𝑠𝑒𝑐 𝜃 > 0, 𝑠𝑒𝑐 𝜃 > 0, 𝑐𝑜𝑡 𝜃 > 0.
In the first quadrant 𝒂𝒍𝒍 the trigonometric functions are
positive.
b) If 𝜋
2 < 𝜃 < 𝜋 then 𝑃(𝑥,𝑦 ) is a point in the second quadrant.
Here 𝑥 < 0 and 𝑦 > 0. ∴ 𝑠𝑖𝑛 𝜃 > 0, 𝑐𝑜𝑠 𝜃 < 0 , 𝑡𝑎𝑛 𝜃 < 0
Page 44
∴ 𝑐𝑜𝑠𝑒𝑐 𝜃 > 0, 𝑠𝑒𝑐 𝜃 < 0, 𝑐𝑜𝑡 𝜃 < 0.
In the second quadrant 𝒔𝒊𝒏 𝜽 (and also 𝒐𝒔𝒆𝒄 𝜽 ) is positive.
c) If 𝜋 < 𝜃 < 3𝜋
2 then 𝑃(𝑥,𝑦 ) is a point in the third quadrant.
Here 𝑥 < 0 and 𝑦 < 0. ∴ 𝑠𝑖𝑛 𝜃 < 0, 𝑐𝑜𝑠 𝜃 < 0, 𝑡𝑎𝑛 𝜃 > 0
∴ 𝑐𝑜𝑠𝑒𝑐 𝜃 < 0, 𝑠𝑒𝑐 𝜃 < 0, 𝑐𝑜𝑡 𝜃 > 0.
In the third quadrant 𝒕𝒂𝒏 𝜽 ( and also 𝒄𝒐𝒕 𝜽 ) is positive.
d) If 3𝜋
2 < 𝜃 < 2𝜋 then 𝑃(𝑥,𝑦) is a point in the
fourth quadrant. Here 𝑥 > 0 and 𝑦 < 0.
∴ 𝑠𝑖𝑛 𝜃 < 0, 𝑐𝑜𝑠 𝜃 > 0 and 𝑡𝑎𝑛 𝜃 < 0.
∴ 𝑐𝑜𝑠𝑒𝑐 𝜃 < 0, 𝑠𝑒𝑐 𝜃 > 0 , 𝑐𝑜𝑡 𝜃 < 0.
In the fourth quadrant 𝒄𝒐𝒔 𝜽 (and also 𝒆𝒄 𝜽 ) is positive.
2. Find the range of values of the trigonometric functions
Solution: Consider a circle of radius 𝑟, centered at the origin
of a Cartesian coordinate system.
Let 𝑃(𝑥, 𝑦) be a point on the circle.
Suppose that the angle made by the
radius 𝑂𝑃 with the positive direction
of the x-axis is 𝜃.
Then 𝑠𝑖𝑛 𝜃 = 𝑦
𝑟 , 𝑐𝑜𝑠 𝜃 =
𝑥
𝑟 ,
𝑡𝑎𝑛 𝜃 = 𝑦
𝑥 ,
𝑐𝑜𝑠𝑒𝑐 𝜃 = 𝑟
𝑦 , 𝑠𝑒𝑐 𝜃 =
𝑟
𝑥 , 𝑐𝑜𝑡 𝜃 =
𝑥
𝑦.
For any position of the point 𝑃(𝑥, 𝑦) on the circle,
− 𝑟 ≤ 𝑦 ≤ 𝑟 ⇒ − 1 ≤ 𝑦/𝑟 ≤ 1 ⇒ − 1 ≤ 𝑠𝑖𝑛𝜃 ≤ 1.
Similarly
− 𝑟 ≤ 𝑥 ≤ 𝑟 ⇒ − 1 ≤ 𝑥/𝑟 ≤ 1 ⇒ − 1 ≤ 𝑐𝑜𝑠𝜃 ≤ 1.
Since − ∞ < 𝑦/𝑥 < ∞ and − ∞ < 𝑥/𝑦 < ∞, we get,
− ∞ < 𝑡𝑎𝑛𝜃 < ∞ and − ∞ < 𝑐𝑜𝑡𝜃 < ∞.
Also, – 𝑟 ≤ 𝑦 ≤ 𝑟 ⇒ 𝑟/𝑦 ≥ 1 or 𝑟/𝑦 ≤ − 1
∴ | 𝑟/𝑦 | ≥ 1. ∴ |𝑠𝑒𝑐𝜃 | ≥ 1.
Page 45
Similarly – 𝑟 ≤ 𝑥 ≤ 𝑟 ⇒ 𝑟/𝑥 ≥ 1 or 𝑟/𝑥 ≤ − 1
∴ | 𝑟/𝑥 | ≥ 1. ∴ |𝑐𝑜𝑠𝑒𝑐𝜃 | ≥ 1.
3. Express the trigonometric ratios of − 𝜃 with those of 𝜃.
Solution: Consider a circle of radius 𝑟 centered at the origin of
the Cartesian coordinate system. Let
𝑃( 𝑥, 𝑦 ) be a point on the circle. Suppose
that the angle made by the radius 𝑂𝑃
with the positive direction of the x-axis is
𝜃. Then
𝑠𝑖𝑛 𝜃 = 𝑦
𝑟 , 𝑐𝑜𝑠 𝜃 =
𝑥
𝑟 , 𝑡𝑎𝑛 𝜃 =
𝑦
𝑥.
Draw 𝑃𝑀 perpendicular to the x-axis and
produce to meet the circle at 𝑄.
Here 𝑄 is the image of 𝑃. ∴ 𝑄 ≡ ( 𝑥, − 𝑦 ). Here angle 𝑄𝑂𝑋 is
− 𝜃.
By definition,
𝑠𝑖𝑛 (− 𝜃) = − 𝑦
𝑟 = −
𝑦
𝑟 = − 𝑠𝑖𝑛 𝜃 ;
𝑐𝑜𝑠 (− 𝜃) = 𝑥
𝑟 = 𝑐𝑜𝑠 𝜃 ;
𝑡𝑎𝑛 (− 𝜃) = − 𝑦
𝑥 = −
𝑦
𝑥 = − 𝑡𝑎𝑛 𝜃.
By writing the reciprocals we get 𝑐𝑜𝑠𝑒𝑐 (− 𝜃) = − 𝑐𝑜𝑠𝑒𝑐 𝜃 ;
𝑠𝑒𝑐 (− 𝜃) = 𝑠𝑒𝑐 𝜃 ; 𝑐𝑜𝑡 (− 𝜃) = − 𝑐𝑜𝑡 𝜃.
4. Prove geometrically 𝑐𝑜𝑠 (𝑥 + 𝑦) = 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 − 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦. Hence
prove that 𝑐𝑜𝑠 (𝑥 − 𝑦) = 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 + 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦
Solution: Consider the unit circle centered at the origin of the Cartesian
coordinate system.
Page 46
Let 𝑃 and 𝑄 be points on the unit circle such that angle 𝑋 = 𝑥 , angle
𝑄𝑂𝑃 = 𝑦 and angle 𝑄𝑂𝑋 = 𝑥 + 𝑦.
Then 𝑃 ≡ (𝑐𝑜𝑠 𝑥, 𝑠𝑖𝑛 𝑥) and 𝑄 ≡ (𝑐𝑜𝑠 (𝑥 + 𝑦), 𝑠𝑖𝑛 (𝑥 + 𝑦)).
Let 𝑅 be a point on the unit circle such that angle 𝑄𝑂𝑋 = − 𝑦.
Then 𝑅 ≡ (𝑐𝑜𝑠 (− 𝑦), 𝑠𝑖𝑛 (− 𝑦)). Let 𝑆 ≡ (1, 0).
In ∆𝑃𝑂𝑅 and ∆𝑄𝑂𝑆, 𝑂𝑃 = 𝑂𝑄 = 𝑂𝑅 = 𝑂𝑆.
Also angle 𝑃𝑂𝑅 = 𝑄𝑂𝑆.
∴ triangles are congruent ⇒ 𝑃𝑅 = 𝑄𝑆 ⇒ 𝑃𝑅2 = 𝑄𝑆2.
⇒ 𝑐𝑜𝑠 𝑥 − 𝑐𝑜𝑠(− 𝑦) 2 + 𝑠𝑖𝑛 𝑥 − 𝑠𝑖𝑛(− 𝑦) 2
= 𝑐𝑜𝑠 (𝑥 + 𝑦) − 1 2 + 𝑠𝑖𝑛 (𝑥 + 𝑦) − 0 2
⇒ 𝑐𝑜𝑠 𝑥 − 𝑐𝑜𝑠 𝑦 2 + 𝑠𝑖𝑛 𝑥 + 𝑠𝑖𝑛 𝑦 2
= 𝑐𝑜𝑠 (𝑥 + 𝑦) − 1 2 + 𝑠𝑖𝑛 (𝑥 + 𝑦) 2
⇒ 𝑐𝑜𝑠2 𝑥 + 𝑐𝑜𝑠2 𝑦 − 2. 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 + 𝑠𝑖𝑛2 𝑥 + 𝑠𝑖𝑛2 𝑦 + 2. 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦
= 𝑐𝑜𝑠2 (𝑥 + 𝑦) + 1 − 2. 𝑐𝑜𝑠 (𝑥 + 𝑦) + 𝑠𝑖𝑛2 (𝑥 + 𝑦)
⇒ 2 − 2. 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 + 2. 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦 = 2 − 2. 𝑐𝑜𝑠 (𝑥 + 𝑦)
⇒ − 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 + 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦 = − 𝑐𝑜𝑠 (𝑥 + 𝑦)
⇒ 𝑐𝑜𝑠 (𝑥 + 𝑦) = 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 − 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦.
By replacing 𝑦 by − 𝑦 we get
𝑐𝑜𝑠 (𝑥 + (−𝑦)) = 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 (−𝑦) − 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 (−𝑦)
⇒ 𝑐𝑜𝑠 (𝑥 − 𝑦) = 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 − 𝑠𝑖𝑛 𝑥. (− 𝑠𝑖𝑛 𝑦)
= 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 + 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦.
5. Express the trigonometric ratios of 90° − 𝜃 with those of 𝜃
Solution: We know that 𝑐𝑜𝑠 (𝑥 + 𝑦) = 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 − 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦.
a) By taking 𝑥 = 90° and 𝑦 = − 𝜃 we get,
𝑐𝑜𝑠 (90° − 𝜃) = 𝑐𝑜𝑠 90°. 𝑐𝑜𝑠 (− 𝜃) − 𝑠𝑖𝑛 90°. 𝑠𝑖𝑛 ( −𝜃)
= (0). 𝑐𝑜𝑠 𝜃 − (1)(−𝑠𝑖𝑛 𝜃) = 𝑠𝑖𝑛 𝜃.
∴ 𝑐𝑜𝑠 (90° − 𝜃) = 𝑠𝑖𝑛 𝜃.
b) Replacing 𝜃 by 90° − 𝜃 in 𝑠𝑖𝑛 𝜃 = 𝑐𝑜𝑠 (90° − 𝜃) we get,
𝑠𝑖𝑛 (90° − 𝜃) = 𝑐𝑜𝑠 90° − (90° − 𝜃) = 𝑐𝑜𝑠 𝜃.
c) 𝑡𝑎𝑛 (90° − 𝜃) = 𝑠𝑖𝑛 (90° −𝜃)
𝑐𝑜𝑠 (90° − 𝜃) =
𝑐𝑜𝑠 𝜃
𝑠𝑖𝑛 𝜃 = 𝑐𝑜𝑡 𝜃.
d) 𝑐𝑜𝑡 (90° − 𝜃) = 𝑐𝑜𝑠 (90° −𝜃)
𝑠𝑖𝑛 (90° − 𝜃) =
𝑠𝑖𝑛 𝜃
𝑐𝑜𝑠 𝜃 = 𝑡𝑎𝑛 𝜃
Page 47
e) 𝑐𝑜𝑠𝑒𝑐 (90° − 𝜃) = 1
𝑠𝑖𝑛 (90° −𝜃) =
1
𝑐𝑜𝑠 𝜃 = 𝑠𝑒𝑐 𝜃.
f) 𝑠𝑒𝑐 (90° − 𝜃) = 1
𝑐𝑜𝑠 (90° −𝜃) =
1
𝑠𝑖𝑛 𝜃 = 𝑐𝑜𝑠𝑒𝑐 𝜃.
6. Express the trigonometric ratios of 90° + 𝜃 with those of 𝜃.
Solution: We know that 𝑐𝑜𝑠 (𝑥 + 𝑦) = 𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 − 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦.
By taking 𝑥 = 90° and 𝑦 = 𝜃 we get,
a) 𝑐𝑜𝑠 (90° + 𝜃) = 𝑐𝑜𝑠 90°. 𝑐𝑜𝑠 𝜃 − 𝑠𝑖𝑛 90°. 𝑠𝑖𝑛 𝜃
= (0). 𝑐𝑜𝑠 𝜃 − (1). 𝑠𝑖𝑛 𝜃 = − 𝑠𝑖𝑛 𝜃.
∴ 𝑐𝑜𝑠 (90° + 𝜃) = − 𝑠𝑖𝑛 𝜃.
b) 𝑠𝑖𝑛 (90° + 𝜃) = 𝑐𝑜𝑠 90° − (90° + 𝜃) = 𝑐𝑜𝑠 (− 𝜃) = 𝑐𝑜𝑠 𝜃.
c) 𝑡𝑎𝑛 (90° + 𝜃) = 𝑠𝑖𝑛 (90° + 𝜃)
𝑐𝑜𝑠 (90° + 𝜃) =
𝑐𝑜𝑠 𝜃
− 𝑠𝑖𝑛 𝜃 = − 𝑐𝑜𝑡 𝜃.
d) 𝑐𝑜𝑡 (90° + 𝜃) = 𝑐𝑜𝑠 (90° + 𝜃)
𝑠𝑖𝑛 (90° + 𝜃) =
− 𝑠𝑖𝑛 𝜃
𝑐𝑜𝑠 𝜃 = − 𝑡𝑎𝑛 𝜃
e) 𝑐𝑜𝑠𝑒𝑐 (90° + 𝜃) = 1
𝑠𝑖𝑛 (90° + 𝜃) =
1
𝑐𝑜𝑠 𝜃 = 𝑠𝑒𝑐 𝜃.
f) 𝑠𝑒𝑐 (90° + 𝜃) = 1
𝑐𝑜𝑠 (90° + 𝜃) =
1
− 𝑠𝑖𝑛 𝜃 = − 𝑐𝑜𝑠𝑒𝑐 𝜃.
7. Find the value of 𝑠𝑖𝑛 18°.
Hence find 𝑐𝑜𝑠 36°.
Solution: Let 𝜃 = 18°. Then 5𝜃 = 90°.
∴ 3𝜃 + 2𝜃 = 90° or 2𝜃 = 90° − 3𝜃.
∴ 𝑠𝑖𝑛 2𝜃 = 𝑠𝑖𝑛 (90° − 3𝜃) = 𝑐𝑜𝑠 3𝜃.
∴ 2𝑠𝑖𝑛 𝜃. 𝑐𝑜𝑠 𝜃 = 4𝑐𝑜𝑠3 𝜃 − 3𝑐𝑜𝑠 𝜃.
Dividing by 𝑐𝑜𝑠 𝜃, we get, 2𝑠𝑖𝑛 𝜃 = 4𝑐𝑜𝑠2 𝜃 − 3
⇒ 2𝑠𝑖𝑛 𝜃 = 4(1 − 𝑠𝑖𝑛2 𝜃) − 3
⇒ 2𝑠𝑖𝑛 𝜃 = 4 − 4𝑠𝑖𝑛2 𝜃 − 3
⇒ 2𝑠𝑖𝑛 𝜃 = 1 − 4𝑠𝑖𝑛2 𝜃
⇒ 4𝑠𝑖𝑛2 𝜃 + 2𝑠𝑖𝑛 𝜃 − 1 = 0
⇒ 𝑠𝑖𝑛 𝜃 = − 2 ± 4 − 4(4)(− 1)
2 × 4 =
− 2 ± 4 + 16
8 =
− 2 ± 20
8
= − 2 ± 2 5
8 =
2(− 1 ± 5)
8 =
− 1 ± 5
4.
Since 𝜃 = 18°, 𝑠𝑖𝑛 𝜃 = 𝑠𝑖𝑛 18° is positive. ∴ 𝑠𝑖𝑛 18° = 5 − 1
4.
By using 𝑐𝑜𝑠 2𝐴 = 1 − 2𝑠𝑖𝑛2𝐴, we get,
Page 48
𝑐𝑜𝑠 36° = 1 − 2𝑠𝑖𝑛2 18° = 1 − 2 5 − 1
4
2
= 1 − 2 5 + 1 − 2 5
16 = 1 −
6 − 2 5
8 =
8 − (6 − 2 5)
8
= 2 + 2 5
8 =
1 + 5
4.
∴ 𝑐𝑜𝑠 36° = 5 + 1
4.
8. Find the value of 𝑠𝑖𝑛 18°.
Hence find 𝑐𝑜𝑠 18°
Solution: Let 𝜃 = 18°. Then 5𝜃 = 90°.
∴ 3𝜃 + 2𝜃 = 90° or 2𝜃 = 90° − 3𝜃.
∴ 𝑠𝑖𝑛 2𝜃 = 𝑠𝑖𝑛 (90° − 3𝜃) = 𝑐𝑜𝑠 3𝜃.
∴ 2𝑠𝑖𝑛 𝜃. 𝑐𝑜𝑠 𝜃 = 4𝑐𝑜𝑠3 𝜃 − 3𝑐𝑜𝑠 𝜃.
Dividing by 𝑐𝑜𝑠 𝜃, we get, 2𝑠𝑖𝑛 𝜃 = 4𝑐𝑜𝑠2 𝜃 − 3
⇒ 2𝑠𝑖𝑛 𝜃 = 4(1 − 𝑠𝑖𝑛2 𝜃) − 3
⇒ 2𝑠𝑖𝑛 𝜃 = 4 − 4𝑠𝑖𝑛2 𝜃 − 3
⇒ 2𝑠𝑖𝑛 𝜃 = 1 − 4𝑠𝑖𝑛2 𝜃
⇒ 4𝑠𝑖𝑛2 𝜃 + 2𝑠𝑖𝑛 𝜃 − 1 = 0
⇒ 𝑠𝑖𝑛 𝜃 = − 2 ± 4 − 4(4)(− 1)
2 × 4 =
− 2 ± 4 + 16
8 =
− 2 ± 20
8
= − 2 ± 2 5
8 =
2(− 1 ± 5)
8 =
− 1 ± 5
4.
Since 𝜃 = 18°, 𝑠𝑖𝑛 𝜃 = 𝑠𝑖𝑛 18° is positive. ∴ 𝑠𝑖𝑛 18° = 5 − 1
4.
By using 𝑐𝑜𝑠 2𝐴 = 1 − 2𝑠𝑖𝑛2𝐴, we get,
𝑐𝑜𝑠 36° = 1 − 2𝑠𝑖𝑛2 18° = 1 − 2 5 − 1
4
2
= 1 − 2 5 + 1 − 2 5
16 = 1 −
6 − 2 5
8 =
8 − (6 − 2 5)
8
= 2 + 2 5
8 =
1 + 5
4.
∴ 𝑐𝑜𝑠 36° = 5 + 1
4.
By using 2𝑐𝑜𝑠2𝐴 = 1 + 𝑐𝑜𝑠 2𝐴, we get,
2𝑐𝑜𝑠218° = 1 + 𝑐𝑜𝑠 36° = 1 + 5 + 1
4 =
4 + 5 + 1
4 =
5 + 5
4
Page 49
∴ 𝑐𝑜𝑠218° = 5 + 5
8
∴ 𝑐𝑜𝑠 18° = 5 + 5
8 =
2(5 + 5)
16 =
10 + 2 5
4.
9. Find the value of 𝑠𝑖𝑛 18°.
Hence find 𝑠𝑖𝑛 36°.
Solution: Let 𝜃 = 18°. Then 5𝜃 = 90°.
∴ 3𝜃 + 2𝜃 = 90° or 2𝜃 = 90° − 3𝜃.
∴ 𝑠𝑖𝑛 2𝜃 = 𝑠𝑖𝑛 (90° − 3𝜃) = 𝑐𝑜𝑠 3𝜃.
∴ 2𝑠𝑖𝑛 𝜃. 𝑐𝑜𝑠 𝜃 = 4𝑐𝑜𝑠3 𝜃 − 3𝑐𝑜𝑠 𝜃.
Dividing by 𝑐𝑜𝑠 𝜃, we get, 2𝑠𝑖𝑛 𝜃 = 4𝑐𝑜𝑠2 𝜃 − 3
⇒ 2𝑠𝑖𝑛 𝜃 = 4(1 − 𝑠𝑖𝑛2 𝜃) − 3
⇒ 2𝑠𝑖𝑛 𝜃 = 4 − 4𝑠𝑖𝑛2 𝜃 − 3
⇒ 2𝑠𝑖𝑛 𝜃 = 1 − 4𝑠𝑖𝑛2 𝜃
⇒ 4𝑠𝑖𝑛2 𝜃 + 2𝑠𝑖𝑛 𝜃 − 1 = 0
⇒ 𝑠𝑖𝑛 𝜃 = − 2 ± 4 − 4(4)(− 1)
2 × 4 =
− 2 ± 4 + 16
8 =
− 2 ± 20
8
= − 2 ± 2 5
8 =
2(− 1 ± 5)
8 =
− 1 ± 5
4.
Since 𝜃 = 18°, 𝑠𝑖𝑛 𝜃 = 𝑠𝑖𝑛 18° is positive. ∴ 𝑠𝑖𝑛 18° = 5 − 1
4.
By using 𝑐𝑜𝑠 2𝐴 = 1 − 2𝑠𝑖𝑛2𝐴, we get,
𝑐𝑜𝑠 36° = 1 − 2𝑠𝑖𝑛2 18° = 1 − 2 5 − 1
4
2
= 1 − 2 5 + 1 − 2 5
16 = 1 −
6 − 2 5
8 =
8 − (6 − 2 5)
8
= 2 + 2 5
8 =
1 + 5
4.
∴ 𝑐𝑜𝑠 36° = 5 + 1
4.
∴ 𝑠𝑖𝑛2 36° = 1 − 𝑐𝑜𝑠2 36° = 1 − 5 + 1
4
2
= 1 − 5 + 1 + 2 5
16
= 1 − 6 + 2 5
16 =
16 − 6 − 2 5
16 =
10 − 2 5
16
∴ 𝑠𝑖𝑛 36° = 10 − 2 5
4.
Page 50
10. Show that 𝑐𝑜𝑠 𝐴 + 𝑐𝑜𝑠 120° + 𝐴 + 𝑐𝑜𝑠 120° − 𝐴 = 0.
Solution: We have
𝑐𝑜𝑠 120° + 𝐴 = 𝑐𝑜𝑠 120°. 𝑐𝑜𝑠 𝐴 − 𝑠𝑖𝑛 120°. 𝑠𝑖𝑛 𝐴 and
𝑐𝑜𝑠 120° − 𝐴 = 𝑐𝑜𝑠 120°. 𝑐𝑜𝑠 𝐴 + 𝑠𝑖𝑛 120°. 𝑠𝑖𝑛 𝐴
∴ 𝑐𝑜𝑠 120° + 𝐴 + 𝑐𝑜𝑠 120° − 𝐴 = 2 𝑐𝑜𝑠 120° 𝑐𝑜𝑠 𝐴
= 2 𝑐𝑜𝑠 180° − 60° 𝑐𝑜𝑠 𝐴
= − 2𝑐𝑜𝑠 60° . 𝑐𝑜𝑠 𝐴 = − 2 1
2 𝑐𝑜𝑠 𝐴 = − 𝑐𝑜𝑠 𝐴.
∴ 𝑐𝑜𝑠 𝐴 + 𝑐𝑜𝑠 120° + 𝐴 + 𝑐𝑜𝑠 120° − 𝐴
= 𝑐𝑜𝑠 𝐴 − 𝑐𝑜𝑠 𝐴 = 0 = 𝑅𝐻𝑆.
11. Prove that 𝑐𝑜𝑠 20° 𝑐𝑜𝑠 40° 𝑐𝑜𝑠 60° 𝑐𝑜𝑠 80° = 1
16.
Solution: 𝐿𝐻𝑆 =1
2 𝑐𝑜𝑠 20° + 40° + 𝑐𝑜𝑠 20° − 40°
1
2. 𝑐𝑜𝑠 80°.
=1
4 𝑐𝑜𝑠 60° + 𝑐𝑜𝑠 − 20° . 𝑐𝑜𝑠 80°
=1
4
1
2 + 𝑐𝑜𝑠 20° . 𝑐𝑜𝑠 80° =
1
4
1 + 2 𝑐𝑜𝑠 20°
2 . 𝑐𝑜𝑠 80°
=1
8 𝑐𝑜𝑠 80° + 2 𝑐𝑜𝑠 80° 𝑐𝑜𝑠 20°
=1
8 𝑐𝑜𝑠 80° + 𝑐𝑜𝑠 80° + 20° + 𝑐𝑜𝑠 80° − 20°
=1
8 𝑐𝑜𝑠 80° + 𝑐𝑜𝑠 100° + 𝑐𝑜𝑠 60°
= 1
8 𝑐𝑜𝑠 80° + 𝑐𝑜𝑠 180° − 80° +
1
2
=1
8 𝑐𝑜𝑠 80° − 𝑐𝑜𝑠 80° +
1
2 =
1
16
II method:
𝑐𝑜𝑠 20° 𝑐𝑜𝑠 40° 𝑐𝑜𝑠 60° 𝑐𝑜𝑠 80°
= 𝑐𝑜𝑠 60° . 𝑐𝑜𝑠 20° 𝑐𝑜𝑠(60° − 20°) 𝑐𝑜𝑠(60° + 20°)
= 𝑐𝑜𝑠 60° . 𝑐𝑜𝑠 20° 𝑐𝑜𝑠2 60° − 𝑠𝑖𝑛2 20°
=1
2. 𝑐𝑜𝑠 20°
1
2
2 − ( 1 − 𝑐𝑜𝑠2 20°)
=1
2. 𝑐𝑜𝑠 20°
1
4 − 1 + 𝑐𝑜𝑠2 20°
=1
2. 𝑐𝑜𝑠 20° 𝑐𝑜𝑠2 20° −
3
4
=1
2. 𝑐𝑜𝑠 20°
4𝑐𝑜𝑠 2 20° − 3
4
Page 51
=1
2.
4𝑐𝑜𝑠 3 20° − 3 𝑐𝑜𝑠 20°
4
= 1
2.
1
4. 𝑐𝑜𝑠 (3 × 20°) =
1
2.
1
4. 𝑐𝑜𝑠 60° =
1
8.
1
2 =
1
16.
12. Show that 𝑠𝑖𝑛 𝜃 =𝑠𝑖𝑛 3𝜃
1 + 2𝑐𝑜𝑠 2𝜃 and hence prove that
𝑠𝑖𝑛15° = 3 − 1
2 2.
Solution: 𝑠𝑖𝑛 3𝜃
1 + 2𝑐𝑜𝑠 2𝜃 =
3𝑠𝑖𝑛 𝜃 − 4𝑠𝑖𝑛 3 𝜃
1 + 2(1 − 2𝑠𝑖𝑛 2 𝜃) =
𝑠𝑖𝑛 𝜃(3 − 4𝑠𝑖𝑛 2 𝜃)
3 − 4𝑠𝑖𝑛 2 𝜃 = 𝑠𝑖𝑛 𝜃.
∴ 𝑠𝑖𝑛 𝜃 =𝑠𝑖𝑛 3𝜃
1 + 2𝑐𝑜𝑠 2𝜃.
By taking 𝜃 = 15° we get,
𝑠𝑖𝑛 15° =𝑠𝑖𝑛 3(15°)
1 + 2𝑐𝑜𝑠 2(15°) =
𝑠𝑖𝑛 45°
1 + 2𝑐𝑜𝑠 30° =
1 2
1 + 2 3 2 =
1
2 1 + 3
= 1
2 3 + 1 =
1
2 3 + 1 . 3 − 1
3 − 1 =
3 − 1
2 3 − 1 =
3 − 1
2 2.
13. Show that 𝑐𝑜𝑠 7𝑥 − 𝑐𝑜𝑠 5𝑥 + 𝑐𝑜𝑠 3𝑥 − 𝑐𝑜𝑠 𝑥
𝑠𝑖𝑛 7𝑥 − 𝑠𝑖𝑛 5𝑥 − 𝑠𝑖𝑛 3𝑥 + 𝑠𝑖𝑛 𝑥 = 𝑐𝑜𝑡 2𝑥.
Solution: 𝑐𝑜𝑠 7𝑥 − 𝑐𝑜𝑠 5𝑥 + 𝑐𝑜𝑠 3𝑥 − 𝑐𝑜𝑠 𝑥
𝑠𝑖𝑛 7𝑥 − 𝑠𝑖𝑛 5𝑥 − 𝑠𝑖𝑛 3𝑥 + 𝑠𝑖𝑛 𝑥 =
𝑐𝑜𝑠 7𝑥 + 𝑐𝑜𝑠 3𝑥 − 𝑐𝑜𝑠 5𝑥 − 𝑐𝑜𝑠 𝑥
𝑠𝑖𝑛 7𝑥 − 𝑠𝑖𝑛 3𝑥 − 𝑠𝑖𝑛 5𝑥 + 𝑠𝑖𝑛 𝑥
=𝑐𝑜𝑠 7𝑥 + 𝑐𝑜𝑠 3𝑥 − (𝑐𝑜𝑠 5𝑥 + 𝑐𝑜𝑠 𝑥)
𝑠𝑖𝑛 7𝑥 − 𝑠𝑖𝑛 3𝑥 − (𝑠𝑖𝑛 5𝑥 − 𝑠𝑖𝑛 𝑥)
=2 𝑐𝑜𝑠
7𝑥 + 3𝑥2
.𝑐𝑜𝑠 7𝑥 − 3𝑥
2 − 2 𝑐𝑜𝑠 5𝑥 + 𝑥
2 .𝑐𝑜𝑠
5𝑥 − 𝑥
2
2 𝑐𝑜𝑠 7𝑥 + 3𝑥
2 .𝑠𝑖𝑛
7𝑥 − 3𝑥
2 − 2 𝑐𝑜𝑠
5𝑥 + 𝑥2
.𝑠𝑖𝑛 5𝑥 − 𝑥
2
=2𝑐𝑜𝑠 5𝑥∙𝑐𝑜𝑠 2𝑥 − 2𝑐𝑜𝑠 3𝑥∙𝑐𝑜𝑠 2𝑥
2𝑐𝑜𝑠 5𝑥∙𝑠𝑖𝑛 2𝑥 − 2𝑐𝑜𝑠 3𝑥∙𝑠𝑖𝑛 2𝑥
=2𝑐𝑜𝑠 2𝑥(𝑐𝑜𝑠 5𝑥 − 𝑐𝑜𝑠 3𝑥)
2𝑠𝑖𝑛 2𝑥(𝑐𝑜𝑠 5𝑥 − 𝑐𝑜𝑠 3𝑥) =
𝑐𝑜𝑠 2𝑥
𝑠𝑖𝑛 2𝑥 = 𝑐𝑜𝑡 2𝑥.
14. Prove that 𝑐𝑜𝑠 𝑥 + 𝑐𝑜𝑠 𝑦 2 + 𝑠𝑖𝑛 𝑥 − 𝑠𝑖𝑛 𝑦 2 = 4𝑐𝑜𝑠2 𝑥 + 𝑦
2 .
Solution: 𝑐𝑜𝑠 𝑥 + 𝑐𝑜𝑠 𝑦 2 + 𝑠𝑖𝑛 𝑥 − 𝑠𝑖𝑛 𝑦 2
= 𝑐𝑜𝑠2𝑥 + 𝑐𝑜𝑠2 𝑦 + 2𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 + 𝑠𝑖𝑛2 𝑥 + 𝑠𝑖𝑛2 𝑦 − 2𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦
= 𝑐𝑜𝑠2𝑥 + 𝑠𝑖𝑛2 𝑥 + 𝑐𝑜𝑠2 𝑦 + 𝑠𝑖𝑛2 𝑦 + 2(𝑐𝑜𝑠 𝑥. 𝑐𝑜𝑠 𝑦 − 𝑠𝑖𝑛 𝑥. 𝑠𝑖𝑛 𝑦)
= 1 + 1 + 2𝑐𝑜𝑠 (𝑥 + 𝑦)
= 2 + 2𝑐𝑜𝑠 (𝑥 + 𝑦) = 2 1 + 𝑐𝑜𝑠 (𝑥 + 𝑦) = 2.2𝑐𝑜𝑠2 𝑥 + 𝑦
2
= 4𝑐𝑜𝑠2 𝑥 + 𝑦
2 .
Page 52
15. If 𝑡𝑎𝑛 𝑥 = − 4
3,
𝜋
2 < 𝑥 < 𝜋 find 𝑠𝑖𝑛
𝑥
2, 𝑐𝑜𝑠
𝑥
2 and 𝑡𝑎𝑛
𝑥
2.
Solution: Let 𝑜𝑝𝑝 = 4 and 𝑎𝑑𝑗 = 3. Then ℎ𝑦𝑝 = 5.
Since 𝑥 is an angle lying in the II quadrant, 𝑐𝑜𝑠 𝑥 is negative.
∴ 𝑐𝑜𝑠 𝑥 = − 𝑎𝑑𝑗
ℎ𝑦𝑝 = −
4
5.
Now 2𝑠𝑖𝑛2 𝑥
2 = 1 − 𝑐𝑜𝑠 𝑥 = 1 − −
4
5 =
5 + 4
5 =
9
5.
∴ 𝑠𝑖𝑛2 𝑥
2 =
9
10. ∴ 𝑠𝑖𝑛
𝑥
2 = ±
3
10.
Since 𝜋
2 < 𝑥 < 𝜋 we get,
𝜋
4 <
𝑥
2 <
𝜋
2.
∴ 𝑠𝑖𝑛 𝑥
2 > 0. ∴ 𝑠𝑖𝑛
𝑥
2 =
3
10.
Now 2𝑐𝑜𝑠2 𝑥
2 = 1 + 𝑐𝑜𝑠 𝑥 = 1 + −
4
5 =
5 − 4
5 =
1
5.
∴ 𝑐𝑜𝑠2 𝑥
2 =
1
10. ∴ 𝑐𝑜𝑠
𝑥
2 = ±
1
10.
Since 𝜋
2 < 𝑥 < 𝜋 we get,
𝜋
4 <
𝑥
2 <
𝜋
2.
∴ 𝑐𝑜𝑠 𝑥
2 > 0. ∴ 𝑐𝑜𝑠
𝑥
2 =
1
10.
Also 𝑡𝑎𝑛 𝑥
2 =
𝑠𝑖𝑛 𝑥
2
𝑐𝑜𝑠 𝑥
2
= 3 10
1 10 = 3.
16. Prove that 𝑠𝑖𝑛2 𝑥 + 𝑠𝑖𝑛2 𝑥 + 2𝜋
3 + 𝑠𝑖𝑛2 𝑥 −
2𝜋
3 =
3
2.
Solution: 𝑠𝑖𝑛2 𝑥 + 𝑠𝑖𝑛2 𝑥 + 2𝜋
3 + 𝑠𝑖𝑛2 𝑥 −
2𝜋
3
= 1 − 𝑐𝑜𝑠 2𝑥
2 +
1 − 𝑐𝑜𝑠 2 𝑥 + 2𝜋
3
2 +
1 − 𝑐𝑜𝑠 2 𝑥 − 2𝜋
3
2
= 1 − 𝑐𝑜𝑠 2𝑥
2 +
1 − 𝑐𝑜𝑠 2𝑥 + 4𝜋
3
2 +
1 − 𝑐𝑜𝑠 2𝑥 − 4𝜋
3
2
= 1 − 𝑐𝑜𝑠 2𝑥 + 1 − 𝑐𝑜𝑠 2𝑥 +
4𝜋
3 + 1 − 𝑐𝑜𝑠 2𝑥 −
4𝜋
3
2
= 3 − 𝑐𝑜𝑠 2𝑥 + 𝑐𝑜𝑠 2𝑥 +
4𝜋
3 + 𝑐𝑜𝑠 2𝑥 −
4𝜋
3
2
= 3 − 𝑐𝑜𝑠 2𝑥 + 2 𝑐𝑜𝑠 2𝑥 .𝑐𝑜𝑠
4𝜋
3
2
{ 𝑐𝑜𝑠 4𝜋
3 = 𝑐𝑜𝑠
3𝜋 + 𝜋
3 = 𝑐𝑜𝑠 𝜋 +
𝜋
3 = − 𝑐𝑜𝑠
𝜋
3 = −
1
2 }
= 3 − 𝑐𝑜𝑠 2𝑥 + 2 𝑐𝑜𝑠 2𝑥 . −
1
2
2 =
3 − 𝑐𝑜𝑠 2𝑥 − 𝑐𝑜𝑠 2𝑥
2 =
3
2.