http://www.ck12.org Chapter 6. Analytic Trigonometry
Concept Problem Revisited
In order to fully identify sin^215 ◦you need to use the power reducing formula.
sin^2 x=^1 −cos 2 2 xsin^215 ◦=^1 −cos 30◦
2 =1
2 −
√ 3
4
Vocabulary
Anidentityis a statement proved to be true once so that it can be used as a substitution in future simplifications and
proofs.
Guided Practice
- Prove the power reducing identity for sine.
sin^2 x=^1 −cos 2 2 x - Simplify the following identity: sin^4 x−cos^4 x.
- What is the period of the following function?
f(x) =sin 2x·cosx+cos 2x·sinx
Answers: - Start with the double angle identity for cosine.
cos 2x=cos^2 x−sin^2 x
cos 2x= ( 1 −sin^2 x)−sin^2 x
cos 2x= 1 −2 sin^2 xThis expression is an equivalent expression to the double angle identity and is often considered an alternate form.
2 sin^2 x= 1 −cos 2x
sin^2 x=^1 −cos 2 2 x- Here are the steps:
sin^4 x−cos^4 x= (sin^2 x−cos^2 x)(sin^2 x+cos^2 x)
=−(cos^2 x−sin^2 x)
=−cos 2x3.f(x) =sin 2x·cosx+cos 2x·sinxsof(x) =sin( 2 x+x) =sin 3x. Sinceb=3 this implies the period is^23 π.