Trigonometry

Trigonometry“Chance favorsonly the prepared only the prepared mind”

Louis Pasteur (1822 - 1895)

1.1 Plane Angle & Angle Measurements

1.2 Solution to Right Triangles

1.3 The Six Trigonometric Functions

1.4 Solution to Oblique Triangles

1.5 Area of Triangles

Trigonometry

1.5 Area of Triangles

1.6 Trigonometric Identities

1.7 Inverse Trigonometric Functions

1.8 Spherical Trigonometry

PLANE ANGLE & ANGLE MEASUREMENTS

A plane angle is determined by rotating a ray (half-line)about its endpoint called vertex.

Conversion Factors:Terminal Side

1 revolution = 360 degrees

= 2π radians

= 400 gradians

= 6400 mils

ANGLE

VERTEX Initial Side

Types of Angles

Q-1 The measure of 2.25 revolutionscounterclockwise is

A. -835º C. -810º

B. 805º D. 810º B. 805º D. 810º Conversion Factors:


= 2π radians

= 400 gradians

= 6400 mils

Q-2 4800 mils is equivalent to__________degrees.

A. 135 C. 235

B. 270 D. 142Conversion Factors:


= 2π radians

= 400 gradians

= 6400 mils

A. degree C. radian

B. mil D. grad

Q-3 An angular unit equivalent to 1/400 of thecircumference of a circle is called:

Conversion Factors:


= 2π radians

= 400 gradians

= 6400 mils

Angle Pairs

Complementary AnglesSupplementary Angles

Explementary Angles

90A B∠ + ∠ =180A B∠ + ∠ =360A B∠ + ∠ =

Q-4 Find the complement of the angle whosesupplement is 152º.

A. 28º C. 118º

B. 62º D. 38º

Q-5 A certain angle has an explement 5 timesthe supplement. Find the angle. [ECE BoardNov.2002]

A. 67.5 degrees C. 135 degrees

B. 108 degrees D. 58.5 degrees

TRIANGLESTRIANGLES

Right Triangles

The Pythagorean Theorem:

“In a right triangle, the square of the length “In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs”

c2 = a2 + b2

Note:

In any triangle, the sum of any two sides must be greater than the third side; otherwise no triangle can be formed.

If, 2 2 2c a b The triangle is right= + →If, 2 2 2

2 2 2

2 2 2

c a b The triangle is right

c a b The triangle is obtuse

c a b The triangle is acute

= + →> + →< + →

Trigonometric Functions

θ = = θ = =

θ = = θ = =

θ = = θ = =

opposite o adjacent asin cot

hypotenuse h opposite o

adjacent a hypotenuse hcos sec

hypotenuse h adjacent a

opposite o hypotenuse htan cscθ = = θ = =opposite o hypotenuse htan csc

adjacent a opposite o

SOHSOHSOHSOH----CAHCAHCAHCAH----TOATOATOATOA

Q-6 The sides of a triangular lot are130 m,180 m and 190 m. This lot is to be divided by aline bisecting the longest side and drawn fromthe opposite vertex. Find the length of this line.

A. 120 m C. 122 m

2 2 212 2

2median side side opposite= + −

A. 120 m C. 122 m

B. 130 m D. 135 m

ALtitude – perpendicular to opposite side (Intersection: ORTHOCENTER)

Angle Bisector – bisects angle (Intersection: INCENTER)

Median – vertex to midpoint of opposite side (Intersection: CENTROID)

Q-7 The angle of elevation of the top of thetower from a point 40 m. from its base is thecomplement of the angle of elevation of thesame tower at a point 120 m. from it. What isthe height of the tower?

A. 59.7 C. 69.3

B. 28.5 D. 47.6

A. 10 C. 25

Q-8 One leg of a right triangle is 20 cm andthe hypotenuse is 10 cm longer that the otherleg. Find the length of the hypotenuse.

B. 15 D. 20

A. 76.31 m C. 73.16 m

Q-9 A man finds the angle of elevation of thetop of a tower to be 30 degrees. He walks 85m nearer the tower and finds its angle ofelevation to be 60 degrees. What is the heightof the tower ? [ECE Board Apr. 1998]

A. 76.31 m C. 73.16 m

B. 73.31 m D. 73.61 m

Oblique Triangles

a b csinA sinB sinC

= =

The Sine Law

sinA sinB sinC

When to use Sine Law:

• Given two angles and any side.

• Given two sides and an angle opposite one of them .

+ −= + − =

+ −= + − =

+ −= + − =

2 2 22 2 2

2 2 22 2 2

2 2 22 2 2

Standard Form : Alternative Form :

b c aa b c 2bcCosA cosA

2bca c b

b a c 2acCosB cosB2ac

a b cc a b 2abcosC cosC

2ab

�

�

�

The Cosine Law

Use the Laws of Cosine if:

Given three sides

Given two sides and their included angle

Q-10 In a triangle, find the side c if angle C= 100° , side b = 20 and side a = 15

A. 28 C. 29A. 28 C. 29

B. 27 D. 26

Q-11 Points A and B 1000 m apart are plotted ona straight highway running east and west. FromA , the bearing of a tower C is 32 degrees W of Nand from B the bearing of C is 26 degrees N ofE . Approximate the shortest distance of tower Cto the highway. [ECE Board Apr. 1998:]

A. 364 m C. 394 m

B. 374 m D. 384 m

Q-12 A PLDT tower and a monument stand on alevel plane . The angles of depression of the topand bottom of the monument viewed from the topof the PLDT tower are 13° and 35° respectively.The height of the tower is 50 m. Find the heightof the monument.

A. 33.51 m C. 47.30 mA. 33.51 m C. 47.30 m

B. 7.58 m D. 30.57 m

Area of Triangles

Q-13 Given a right triangle ABC. Angle C is theright angle. BC = 4 and the altitude to thehypotenuse is 1 unit. Find the area of thetriangle. ECE Board Apr.2001:

A. 2.0654 sq. u. C.1.0654 sq. u.A. 2.0654 sq. u. C.1.0654 sq. u.

B. 3.0654 sq. u. D.4.0654 sq. u.

Q-14 In a given triangle ABC, the angle C is34°, side a is 29 cm, and side b is 40 cm.Solve for the area of the triangle.

A. 324.332 cm2 C. 317.15 cm2

B. 344.146 cm2 D. 343.44 cm2B. 344.146 cm2 D. 343.44 cm2

Q-15 A right triangle is inscribed in a circle suchthat one side of the triangle is the diameter of acircle. If one of the acute angles of the trianglemeasures 60 degrees and the side opposite thatangle has length 15, what is the area of thecircle? ECE Board Nov. 2002

A. 175.15 C. 235.62

B. 223.73 D. 228.61

Q-16 The sides of a triangle are 8 cm , 10 cm,and 14 cm. Determine the radius of theinscribed and circumscribing circle.

A. 3.45, 7.14 C. 2.45, 8.14

B. 2.45, 7.14 D. 3.45, 8.14

Q-17 Two triangles have equal bases. The altitudeof one triangle is 3 cm more than its base whilethe altitude of the other is 3 cm less than its base.Find the length of the longer altitude if the areas ofthe triangle differ by 21 square centimeters.

A. 10 C. 14A. 10 C. 14

B. 20 D. 15

Trigonometric Identities

2 2

R e c i p r o c a l r e l a t i o n :

1 1 1s i n u c o s u t a n u

c s c u s e c u c o t u

Q u o t i e n t r e l a t i o n

s i n ut a n u

c o s u

P y t h a g o r e a n r e l a t i o n

s i n u c o s u 1

= = =

=

+ =

� � �

�

�

( )( )

( )

A d d i t i o n & s u b t r a c t i o n f o r m u l a

s i n u v s i n u c o s v c o s u s i n v

c o s u v c o s u c o s v s i n u s i n v

t a n u v

± = ±

± =

± =

m

�

�

�

2

2

t a n u t a n v1 t a n u t a n v

D o u b l e A n g l e f o r m u l a :

s i n 2 u 2 s i n u c o s u

c o s 2 u 2 c o s u 1

2 t a n ut a n 2 u

1 t a n u

±

== −

=−

m

�

�

�

Inverse Trigonometric Functions

The Inverse Sine Function

y = arc sin x iff sin y = x

The Inverse Cosine Function

y = arc cos x iff cos y = x

The Inverse Tangent Function

y = arc tan x iff tan y = x

Q-18 If sec (2x-3) = 1 / sin (5x-9), determinethe angle x in degrees

A. 12.56 deg C. 18.57 deg

B. 14.57deg D. 10.18 degB. 14.57deg D. 10.18 deg

( )

( )( )

sin cos 90

cos sin(90 )

tan cot 90

θ θ

θ θθ θ

= −

= −

= −

o

o

o

COFUNCTION RELATIONS1

sec2sin13

1 1

cos 2 sin13sin13 cos 2

AA

A AA A

cofunction

=

=

=

SOLUTION:

( )( )

sec csc 90

csc sec 90

θ θ

θ θ

= −

= −

o

o

( )sin13 sin 90 2

13 90 2

6

cofunction

A A

A A

A

= −= −

=

Q-19 ECE Board Nov.2003

Simplify the expression

4 cos y sin y (1 – 2 sin2y).

A. sec 2y C. tan 4yA. sec 2y C. tan 4y

B. cos 2y D. sin 4y

2

2 2 2 2

sin 2 2sin cos

2 tantan 2

1 tan

cos 2 cos sin 1 2sin 2cos 1

θ θ θθθθ

θ θ θ θ θ

=

=−

= − = − = −

( )( )( )

24cos sin 1 2siny y y−

( )( )22 2sin cos 1 2sin

2sin 2 cos 2

sin 4

y y y

y y

y

= −

==

Q-20 ECE Board Nov. 1996:

If sin A = 2.511x , cos A = 3.06x and sin 2A= 3.939x , find the value of x?

A. 0.265 C. 0.562A. 0.265 C. 0.562

B. 0.256 D. 0.625

Q-21

Solve for x if tan 3x = 5 tanx

A. 20.705° C. 15.705°

° °B. 30.705° D. 35.705°

3

2

3

2

3 3

3tan tantan 3

1 3tantan 3 5 tan

3tan tan5 tan

1 3tan

3tan tan 5 tan 15tan

x x

x xx

x

x x x x

θ θθθ

−=−

=− =

−− = −

=2

2

14 tan 2 tan

2tan

1420.705

x x

x

x

=

=

=

Q-22 If arctan2x + arctan3x = 45 degrees,what is the value of x?ECE Nov. 2003

A. 1/6 C.1/5

B. 1/3 D.1/4

( ) ( )

( )( )

arctan 2 arctan 3 45

, tan 2 ; tan 3

arctan tan arctan tan 45

45

tan 45

tan tan 45

tan tantan 45

1 tan tan

x x

let A x B x

SUBSTITUTE

A B

A B

A B

A B

A B

A B

+ == =

+ =+ =

+ =

+ =+ =

−

( )( )2

tan 451 tan tan

2 31

1 2 3

6 5 1 0

0.1666 & 1

A BSUBSTITUTE

x x

x x

x x

x

=−

+ =−

+ − == −

Spherical Trigonometry

The study of properties of spherical triangles and their measurements.

The TerrestrialSphere

1minute of arc 1nautical mile=

Conversion Factors

1minute of arc 1nautical mile

1nautical mile 6080 ft.

1nautical mile 1.1516 statue mile

1 statue mile 5280 ft.

1knot 1nautical mile per hour

=====

Spherical TriangleA spherical triangle is the triangle enclosed by arcs of three great circles of a sphere.

Sum of Three vertex angle :

A B C 180

A B C 540

Sum of any two sides :

b c a

a c b

a b c

+ + > °+ + < °

+ >+ >+ >

①

②

( )

( )

a b c

Sum of three sides :

0 a b c 360

Spherical Excess :

E A B C 180

Spherical Defect :

D 360 a b c

+ >

° < + + < °

= + + − °

= ° − + +

③

④

⑤

sin sin sin

sin sin sin

a b c

A B C= =

SPHERICAL TRIANGLES:

Law of sines:

Law of cosines (FOR SIDES):

cos cos cos sin sin cos



a b c b c A

b a c a c B

c a b a b C

= += += +

Law of cosines (FOR SIDES):

Q-23 A spherical triangle ABC has sides a =50°, c = 80°, and an angle C = 90°. Findthe third side “b” of the triangle in degrees.

A. 75.33 degrees C. 74.33 degrees

B. 77.25 degrees D. 73.44 degreesB. 77.25 degrees D. 73.44 degrees

( ) ( ) ( ) ( )cos cos cos sin sin cos

cos 80 cos 50 cos sin 50 sin cos 90

0.1736 0.6

74.3

428cos

3

0

c a b a b C

b

b b

b

= += +

== +

74.33b =

Q-24 Given an isosceles triangle with angleA=B=64 degrees, and side b=81 degrees .What is the value of angle C?

A. C.144 26 'o 120 15'oA. C.

B. D.

144 26 'o

135 10'o

120 15'o

150 25'o




A B C B C a

B A C A C b

C A B A B c

= − += − += − +

COSINE LAW FOR ANGLES:

Q-24 Given an isosceles triangle with angleA=B=64 degrees, and side b=81 degrees .What is the value of angle C?

A. C.144 26 'o 120 15'oA. C.

B. D.

144 26 'o

135 10'o

120 15'o

150 25'o




A B C B C a

B A C A C b

C A B A B c

= − += − += − +

COSINE LAW FOR ANGLES:


cos64 cos(64)cos sin 64sin cos81

B A C A C b

C C

= − += − +

( )

cos64 cos(64)cos sin 64sin cos81

0.4384 0.4384cos 0.1406sin

0.4384 0.4384cos(144 26') 0.1406sin 144 26'

0.4384 0.4384

C C

C C

SUBSTITUTE

= − += − +

= − +

=

o o

POP QUIZ…POP QUIZ…

1. Three circles of radii 3,4, and 5 inches, respectively are tangent to each other extremely. Find the largest angle of a triangle formed by joining the centers.

A. 72.6 deg

B. 75.1 deg

C. 73.4 deg

D. 73.3 deg

2. If sinA= 4/5 and sinB= 7/25, what is sin(A+B) if A is in the 3rd quadrant and B is in the 2nd quadrant?

A. -3/5

B. 4/5

C. 3/5

D. 2/5

3. Find the value of (1 + i)12 .

A. 64

B. -64

C. 64iC. 64i

D. -64i

4. The altitude of the sides of a triangle intersect at the point known as

A. Incenter

B. CircumcenterB. Circumcenter

C. Orthocenter

D. Centroid

5. A triangle inscribed in a given triangle whose vertices are the feet of the three perpendiculars to the sides from the same point inside the given triangle.

A. Pedal triangle

B. Scalene triangleB. Scalene triangle

C. Escribed triangle

D. Egyptian triangle

6. The angle of inclination of ascend of a road having 8.25 % grade is ____ degrees.

A. 4.72

B. 4.27B. 4.27

C. 5.12

D. 1.86

7. The sides of a triangle are 8, 10, and 14. Determine the radius of the inscribed circle.

A. 18.9

B. 19.8

C. 17.9

D. 16.9

8. From the top of the 100-ft-tall building a

man observes a car moving toward the

building. If the angle of depression of the

car changes from 22° to 46° during the

period of observation, how far does the car

travel?travel?

A. 120 C. 171

B. 151 D. 180

9. Calculate the angle of elevation of the

line of sight of a person whose eye is 1.7

m above the ground, and is looking at the

top of a tree which is 27.5 m away on

level ground and 18.6 m high.

A. 30 degreesA. 30 degrees

B. 12 degrees

C. 25 degrees

D. 32 degrees

10. If z varies directly as x and inversely as

y, find the percentage change in z if x

increases by 20% and y increases by 25%.

A. z decreases by 5%

B. z decreases by 80%B. z decreases by 80%

C. z decreases by 4%

D. z decreases by 6.25%

Check Time!!!Check Time!!!

1. Three circles of radii 3,4, and 5 inches, respectively are tangent to each other extremely. Find the largest angle of a triangle formed by joining the centers.

A. 72.6 deg

B. 75.1 deg

C. 73.4 deg

D. 73.3 deg

2. If sinA= 4/5 and sinB= 7/25, what is sin(A+B) if A is in the 3rd quadrant and B is in the 2nd quadrant?

A. -3/5

B. 4/5

C. 3/5

D. 2/5

5. A triangle inscribed in a given triangle whose vertices are the feet of the three perpendiculars to the sides from the same point inside the given triangle.

A. Pedal triangle

B. Scalene triangleB. Scalene triangle

C. Escribed triangle

D. Egyptian triangle

6. The angle of inclination of ascend of a road having 8.25 % grade is ____ degrees.

A. 4.72

B. 4.27B. 4.27

C. 5.12

D. 1.86

7. The sides of a triangle are 8, 10, and 14. Determine the radius of the inscribed circle.

A. 18.9

B. 19.8

C. 17.9

D. 16.9

8. From the top of the 100-ft-tall building a

man observes a car moving toward the

building. If the angle of depression of the

car changes from 22° to 46° during the

period of observation, how far does the car

travel?travel?

A. 120 C. 171

B. 151 D. 180

9. Calculate the angle of elevation of the

line of sight of a person whose eye is 1.7

m above the ground, and is looking at

the top of a tree which is 27.5 m away

on level ground and 18.6 m high.

A. 30 degreesA. 30 degrees

B. 12 degrees

C. 25 degrees

D. 32 degrees

10. If z varies directly as x and inversely as

y, find the percentage change in z if x

increases by 20% and y increases by 25%.

A. z decreases by 5%

B. z decreases by 80%B. z decreases by 80%

C. z decreases by 4%

D. z decreases by 6.25%

Trigonometry

Documents

right angle

angle of elevation

angle intersection

right c

included angle q

certain angle

bearing of c

angle conversion factors