Sum and Difference Identities Classify each statement as true or false, then explain your reasoning. 1) sin(90) = sin(90) + sin(0) 3) sin(30) + sin(60) = sin(90) 2) cos(90) = cos(90) + cos(0) 4) tan(45) + tan(45) = tan(90) From these examples, recognize that the distributive property does NOT apply when using the trigonometric functions. …So what rules do apply? Sum and Difference Identities sin ( A +B )=sin A cos B+cos A sin B tan ( A +B )= tan A +tan B 1−tan A tan B sin ( A−B )=sin A cos B−cos A sin B cos ( A +B )=cos A cos B−sin A sin B tan ( A−B )= tan A −tan B 1 +tan A tan B cos ( A−B )=cos A cos B +sin A sin B A. Demonstrate Demonstrate that each identity works by substituting the given values for A and B. Sample: cos ( A +B )=cos A cos B−sin A sin B , when A = π 3 and B= π 6 Write the identity: cos ( A +B )=cos A cos B−sin A sin B Substitute: cos ( π 3 + π 6 )=cos π 3 cos π 6 −sin π 3 sin π 6 Evaluate: cos ( π 2 ) = ( 1 2 ) ⋅ ( √ 3 2 ) − ( √ 3 2 ) ⋅ ( 1 2 ) Simplify: 0= √ 3 4 − √ 3 4 0 = 0 5) cos ( A−B )=cos A cos B +sin A sin B , when A = π 3 and B= π 6 page 1 of 10
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Sum and Difference Identities Classify each statement as true or false, then explain your reasoning.
From these examples, recognize that the distributive property does NOT apply when using the trigonometric functions.
…So what rules do apply?
Sum and Difference Identities
sin( A+B )=sin A cosB+cos A sin Btan(A+B)=tan A+ tanB
1− tan A tan Bsin( A−B)=sin A cosB−cos A sin B
cos (A+B)=cosA cosB−sin A sin Btan(A−B )= tan A−tanB
1+ tan A tan Bcos (A−B )=cos A cosB+sin A sin B
A. Demonstrate Demonstrate that each identity works by substituting the given values for A and B.
Sample: cos (A+B)=cosA cosB−sin A sin B , when A= π3 and B=
π6
Write the identity: cos (A+B)=cosA cos B−sin A sin B
Substitute: cos ( π3 +π6 )=cos
π3 cos
π6−sin
π3 sin
π6
Evaluate: cos (π2 )=( 12 )⋅( √3
2 )−( √32 )⋅( 12 )
Simplify: 0=√34− √3
4
0 = 0
5) cos (A−B )=cos A cosB+sin A sin B , when A= π3 and B=
π6
page 1 of 10
Sum and Difference Identities
6) sin( A+B )=sin A cosB+cos A sin B , when A= π3 and B=
π6
7) sin( A−B)=sin A cosB−cos A sin B , when A= π3 and B=
π6
8) tan(A−B )=tan A−tanB1+ tan A tan B , when A= π
3 and B=π6
9) tan(A+B)=tan A+ tanB1− tan A tan B , when A= π
3 and B=π6
page 2 of 10
Sum and Difference Identities
B. Find Find the exact values for the trigonometric functions at 105 and 15 by writing the angles as a sum or difference of other special values (such as 60 and 45).
Sample: cos (105° )=cos (60°+45 ° )=cos60 °cos 45 °−sin 60 ° sin 45 °
=12⋅√22
−√32
⋅√22
¿ √24 −√6
4 =√2−√64
10) sin(105 ° )=sin(60°+45 ° )
11) tan(105° )= tan(60°+45 °)
12) cos (15° )=cos (60 °−45 ° )
13) sin(15 ° )=sin(60 °−45° )
page 3 of 10
Sum and Difference Identities
14) tan (15° )= tan(60 °−45 ° )
C. Find Use the sum and difference identities to evaluate functions of other rotation angles.
R and S are rotation angles (between 0 and 90) whose terminal sides
intersect the unit circle at (725 ,
2425 ) and (
35 ,
45 ), respectively.
15) Write down the sine, cosine and tangent values for each angle.
sin R = cos R = tan R =
sin S = cos S = tan S =
16) With a calculator in degree mode, solve for the measures of R and S. Then use these values to estimate R+S and RS.R = R+S =
S = RS =
17) Use the sum and difference identities to find the exact values for the trigonometric functions for R+S and RS.sin(R−S )= sin(R+S )=
cos (R−S )= cos (R+S )=
18) How can the answers to #16 be used to check the answers to #17 (using a calculator)?
page 4 of 10
(1, 0)
SR
54
53 ,
2524
257 ,
R+S
RS
Sum and Difference Identities
D. Prove Use the sum and difference identities to verify (or prove) other identities.
Sample: Verify that sin(π2−x )=cos x
sin( π2−x ) =sin π2 cos x−cosπ2 sin x (Sum/Difference Identity)
=(1)cos x−(0)sin x (Evaluate/Substitute)=cos x (Simplify)
19) Prove cos (π2−x )=sin x
20) Prove sec(π2−x )=csc x
21) Prove cos (−x )=cos x Hint: Rewrite x as “0 – x”
22) Prove Hint: Rewrite x as “0 – x”
23) Prove tan (−x )=− tan x Hint: Rewrite x as “0 – x”
page 5 of 10
Sum and Difference Identities KEYClassify each statement as true or false, then explain your reasoning.
FALSE 2) cos(90) = cos(90) + cos(0) FALSE 4) tan(45) + tan(45) = tan(90)No, 0 0 + 1 No, Left side = 2, right side is undefined
From these examples, recognize that the distributive property does NOT apply when using the trigonometric functions.
…So what rules do apply?
Sum and Difference Identities
sin( A+B )=sin A cosB+cos A sin Btan(A+B)=tan A+ tanB
1− tan A tan Bsin( A−B)=sin A cosB−cos A sin B
cos (A+B)=cosA cosB−sin A sin Btan(A−B )= tan A−tanB
1+ tan A tan Bcos (A−B )=cos A cosB+sin A sin B
A. Demonstrate Demonstrate that each identity works by substituting the given values for A and B.
Sample: cos (A+B)=cosA cosB−sin A sin B , when A= π3 and B=
π6
Write the identity: cos (A+B)=cosA cos B−sin A sin B
Substitute: cos ( π3 +π6 )=cos
π3 cos
π6−sin
π3 sin
π6
Evaluate: cos (π2 )=( 12 )⋅( √3
2 )−( √32 )⋅( 12 )
Simplify: 0=√34− √3
4
0 = 0
5) cos (A−B )=cos A cosB+sin A sin B , when A= π3 and B=
π6
page 6 of 10
Sum and Difference Identities KEY
6) sin( A+B )=sin A cosB+cos A sin B , when A= π3 and B=
π6
7) sin( A−B)=sin A cosB−cos A sin B , when A= π3 and B=
π6
8) tan(A−B )=tan A−tanB1+ tan A tan B , when A= π
3 and B=π6
9) tan(A+B)=tan A+ tanB1− tan A tan B , when A= π
3 and B=π6
(Both sides of the equation are undefined.)
page 7 of 10
Sum and Difference Identities KEY
B. Find Find the exact values for the trigonometric functions at 105 and 15 by writing the angles as a sum or difference of other special values (such as 60 and 45).
Sample: cos (105° )=cos (60°+45 ° )=cos60 °cos 45 °−sin 60 ° sin 45 °
=12⋅√22
−√32
⋅√22
¿ √24 −√6
4 =√2−√64
10) sin(105 ° )=sin(60°+45 ° )
11) tan(105° )= tan(60°+45 °)
12) cos (15° )=cos (60 °−45 ° )
13) sin(15 ° )=sin(60 °−45° )
page 8 of 10
Sum and Difference Identities KEY
14) tan (15° )= tan(60 °−45 ° )
C. Find Use the sum and difference identities to evaluate functions of other rotation angles.
(1, 0)
SR
54
53 ,
2524
257 ,
R+S
RS
R and S are rotation angles (between 0 and 90) whose terminal sides
intersect the unit circle at (725 ,
2425 ) and (
35 ,
45 ), respectively.
15) Write down the sine, cosine and tangent values for each angle.
sin R = 24/25 cos R = 7/25 tan R = 24/7
sin S = 4/5 cos S = 3/5 tan S = 4/3
16) With a calculator in degree mode, solve for the measures of R and S. Then use these values to estimate R+S and RS.R = 73.74 R+S = 126.87
S = 53.13 RS = 20.61
17) Use the sum and difference identities to find the exact values for the trigonometric functions for R+S and RS.sin(R−S )= sinR cosS cosR sinS sin(R+S )= sinR cosS + cosR sinS
cos (R−S )= cosR cosS + sinR sinS cos (R+S )= cosR cosS sinR sinS
18) How can the answers to #16 be used to check the answers to #17 (using a calculator)?sin(126.87) 0.8 = 4/5. cos(126.87) -0.6 = -3/5. sin(20.61) 0.352 = 44/125. cos(20.61) 0.936 = 117/125. More exact values could be found using the store key for angles R and S, i.e. sin-1(24/25) ENTER 73.73979529 STO ALPHA R and do the same for S, then use sin(R + S).
page 9 of 10
Sum and Difference Identities KEY
D. Prove Use the sum and difference identities to verify (or prove) other identities.
Sample: Verify that sin(π2−x )=cos x
sin( π2−x ) =sin π2 cos x−cosπ2 sin x (Sum/Difference Identity)
=(1)cos x−(0)sin x (Evaluate/Substitute)=cos x (Simplify)
19) Prove cos (π2−x )=sin x
20) Prove sec(π2−x )=csc x
21) Prove cos (−x )=cos x Hint: Rewrite x as “0 – x”
22) Prove Hint: Rewrite x as “0 – x”
23) Prove tan (−x )=− tan x Hint: Rewrite x as “0 – x”