http://www.ck12.org Chapter 6. Analytic Trigonometry
Guided Practice
- Prove the sine of a difference identity.
- Use a sum or difference identity to find an exact value of cot(^512 π).
- Prove the following identity:
sinsin((xx−+yy))=tantanxx−+tantanyy
Answers:
- Start with the cofunction identity and then distribute and work out the cosine of a sum and cofunction identities.
sin(α−β) =cos(π
2 −(α−β))
=cos((π
2 −α)
+β)
=cos(π
2 −α)
cosβ−sin(π
2 −α)
sinβ
=sinαcosβ−cosαsinβ2.Start with the definition of cotangent as the inverse of tangent.
cot( 5 π
12)
=tan(^15 π
12)
=tan( 9 π^1
12 −^412 π)
=tan( 1351 ◦− 60 ◦)=^1 +tan 135◦tan 60◦
tan 135◦−tan 60◦
=^1 +(−^1 )·√ 3
(− 1 )−√ 3
= (^1 −
√
3 )
(− 1 −√ 3 )
= (^1 −
√ 3 ) 2
(− 1 +√ 3 )·( 1 −√ 3 )
=(^1 −
√ 3 ) 2
−( 1 − 3 )
=(^1 −
√
3 )^2
2
- Here are the steps: