🌊 Trigonometric Functions
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Trigonometric Identities

Prove and apply Pythagorean, reciprocal, and double-angle identities.

Trigonometric identities are equations true for all valid values of the variable. They are used to simplify expressions, solve equations, and prove other identities. The Pythagorean identity is the most fundamental, derived directly from the unit circle definition.

sin⁡2θ+cos⁡2θ=1  ⟹  tan⁡2θ+1=sec⁡2θ  ⟹  1+cot⁡2θ=csc⁡2θ\sin^2\theta + \cos^2\theta = 1 \implies \tan^2\theta + 1 = \sec^2\theta \implies 1 + \cot^2\theta = \csc^2\theta
sin⁡(2θ)=2sin⁡θcos⁡θcos⁡(2θ)=cos⁡2θ−sin⁡2θ\sin(2\theta) = 2\sin\theta\cos\theta \qquad \cos(2\theta) = \cos^2\theta - \sin^2\theta
cos⁡(2θ)=2cos⁡2θ−1=1−2sin⁡2θ\cos(2\theta) = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta

Reciprocal Identities

csc θ = 1/sin θsec θ = 1/cos θcot θ = 1/tan θ = cos θ/sin θ

Quotient Identities

tan θ = sin θ/cos θcot θ = cos θ/sin θ

These six identities, combined with the Pythagorean identities, form the toolkit for simplifying most trig expressions.

🔍Proving an Identity
Prove: sin²θ/(1 − cos θ) = 1 + cos θ. Left side: (1−cos²θ)/(1−cos θ) = (1−cos θ)(1+cos θ)/(1−cos θ) = 1 + cos θ = Right side. ✓
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Remember This!

When proving identities, work on one side only (usually the more complex side). Convert everything to sine and cosine when stuck. Never "cross the equals sign" during a proof.

✏️ Try It!

Using the double-angle formula, which is equivalent to sin(2θ)?

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