6.3. Sum and Difference Identities http://www.ck12.org
Prove the cosine of a sum identity.
Solution: Start with the cosine of a difference and make a substitution. Then use the odd-even identity.
cosαcosβ+sinαsinβ=cos(α−β)
Letγ=−β
cosαcos(−γ)+sinαsin(−γ) =cos(α+γ)
cosαcosγ−sinαsinγ=cos(α+γ)Example B
Find the exact value of tan 15◦without using a calculator.
Solution:tan 15◦=tan( 45 ◦− 30 ◦) = 1 tan 45+tan 45◦−◦tan 30tan 30◦◦=^1 −
√ 3
3
1 + 1 ·√ 3
3=^3 −
√ 3
3 +√ 3
A final solution will not have a radical in the denominator. In this case multiplying through by the conjugate of the
denominator will eliminate the radical. This technique is very common in PreCalculus and Calculus.
=(^3 −
√ 3 )·( 3 −√ 3 )
( 3 +
√
3 )·( 3 −
√
3 )
=(^3 −
√ 3 ) 2
9 − 3
=(^3 −
√ 3 ) 2
6
Example C
Evaluate the expression exactly without using a calculator.
cos 50◦cos 5◦+sin 50◦sin 5◦
Solution: Once you know the general form of the sum and difference identities then you will recognize this as
cosine of a difference.
cos 50◦cos 5◦+sin 50◦sin 5◦=cos( 50 ◦− 5 ◦) =cos 45◦=
√
22
Concept Problem Revisited
In order to evaluate sin 15◦and sin 75◦exactly without a calculator, you need to use the sine of a difference and sine
of a sum.
sin( 45 ◦− 30 ◦) =sin 45◦cos 30◦−cos 45◦sin 30◦=√ 2
2 ·
√ 3
2 −
√ 2
2 ·
1
2 =
√ 6 −√ 2
4
sin( 45 ◦+ 30 ◦) =sin 45◦cos 30◦+cos 45◦sin 30◦=√ 2
2 ·
√ 3
2 +
√ 2
2 ·
1
2 =
√ 6 +√ 2
4
Vocabulary
TheGreek lettersused in this concept refer to unknown angles. They areα-alpha,β-beta,θ-theta,γ-gamma.
The symbol±is short hand for “plus or minus.” The symbol∓is shorthand for “minus or plus.” The order is
important because for cosine of a sum, the negative sign is used on the other side of the identity. This is the opposite
of sine of a sum, where a positive sign is used on the other side of the identity.