Transforming Trigonometric Expressions
Big Idea: Transforming trigonometric expressions means rewriting them in equivalent but simpler or more useful forms using trigonometric identities. This is like factoring in algebra — the expression looks different but has the same value.
🔑 The Essential Identities Toolkit
Before we can transform expressions, we need our toolkit of fundamental identities. Think of these as the "rules of the game."
1. Pythagorean Identities
These come directly from the unit circle and the Pythagorean theorem:
$$ \sin^2\theta + \cos^2\theta = 1 $$
$$ \tan^2\theta + 1 = \sec^2\theta $$
$$ 1 + \cot^2\theta = \csc^2\theta $$
Tip: The first identity is the "mother" identity — the other two are derived from it by dividing by $\cos^2\theta$ or $\sin^2\theta$.
2. Reciprocal & Quotient Identities
Reciprocal:
$ \csc\theta = \dfrac{1}{\sin\theta} $, $ \sec\theta = \dfrac{1}{\cos\theta} $, $ \cot\theta = \dfrac{1}{\tan\theta} $
Quotient:
$ \tan\theta = \dfrac{\sin\theta}{\cos\theta} $
$ \cot\theta = \dfrac{\cos\theta}{\sin\theta} $