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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 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”
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Sum and Difference Identities KEYClassify each statement as true or false, then explain your reasoning.
TRUE 1) sin(90) = sin(90) + sin(0) FALSE 3) sin(30) + sin(60) = sin(90)Yes, 1 = 1 + 0 No, ½ + /2 1
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”
page 10 of 10
