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Equivalent Representations of Trigonometric Functions

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

Two trig expressions are equivalent if they agree at every input. The most-used identities are the Pythagorean identity (sin⁡2+cos⁡2=1\sin^2 + \cos^2 = 1), the quotient identity (tan⁡=sin⁡/cos⁡\tan = \sin/\cos), and the reciprocal identities. Together they let you rewrite almost any trig expression in any equivalent form — and they let you compute the other trig values from one known value and a quadrant.

Pythagorean identity (three forms)

Primary
sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1
Divide by cos²
1+tan⁡2θ=sec⁡2θ1 + \tan^2\theta = \sec^2\theta
Divide by sin²
1+cot⁡2θ=csc⁡2θ1 + \cot^2\theta = \csc^2\theta

Quotient and reciprocal identities

Quotient
tan⁡θ=sin⁡θcos⁡θ,  cot⁡θ=cos⁡θsin⁡θ\tan\theta = \tfrac{\sin\theta}{\cos\theta}, \;\cot\theta = \tfrac{\cos\theta}{\sin\theta}
Reciprocal
sec⁡θ=1cos⁡θ,  csc⁡θ=1sin⁡θ\sec\theta = \tfrac{1}{\cos\theta}, \;\csc\theta = \tfrac{1}{\sin\theta}
AP Tip: If you know sin and the quadrant, the Pythagorean identity gives cos (up to sign), and the quadrant fixes the sign.
Type 1

Pythagorean identity

Square the known value and subtract from 1 to get the squared other trig. Take a square root and attach the quadrant's sign.

Example 1
If sin⁡θ=3/5\sin\theta = 3/5 and θ\theta is in Q2, find cos⁡θ\cos\theta. (A) 4/54/5 (B) −4/5-4/5 (C) 3/43/4 (D) −3/4-3/4

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Type 2

Quotient and reciprocal identities

Substitute the appropriate identity into the expression and simplify.

Example 2
Simplify sin⁡θcos⁡θ⋅cos⁡θ\dfrac{\sin\theta}{\cos\theta} \cdot \cos\theta. (A) sin⁡θ\sin\theta (B) cos⁡θ\cos\theta (C) tan⁡θ\tan\theta (D) 11

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Type 3

Verify or simplify

Apply identities to one side of the equation until it matches the other.

Example 3
Simplify sec⁡2θ−1\sec^2\theta - 1. (A) tan⁡2θ\tan^2\theta (B) cot⁡2θ\cot^2\theta (C) cos⁡2θ\cos^2\theta (D) −sin⁡2θ-\sin^2\theta

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Equivalent Representations of Trigonometric Functions | AP Precalculus — Mathfolis