AP®︎ Precalculus: Topic 3.12 Practice Test

Prepare for your quiz, test, or the AP exam with focused practice questions on Topic 3.12 of AP Precalculus – Equivalent Representations of Trigonometric Functions.

Questions List

Topic 3.12

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Question 1 Easy

Which of the following equations is true for all xx in its domain?

Asin⁡x+cos⁡x=1\sin x+\cos x=1
B1+tan⁡2x=sec⁡2x1+\tan^2x=\sec^2x
Csin⁡(2x)=2sin⁡x\sin(2x)=2\sin x
D1+cot⁡2x=sec⁡2x1+\cot^2x=\sec^2x

What You’re Being Tested On:

Explore the learning objectives taken directly from the College Board’s AP® Precalculus Curriculum. Ensure you’re prepared for the exact topics covered on the AP® exam, in-class tests, and quizzes, and gain confidence in your mastery of the material.

Topic 3.12: Equivalent Representations of Trigonometric Functions

Learning Objective: 3.12.A

Rewrite trigonometric expressions in equivalent forms with the Pythagorean identity.

Essential Knowledge: 3.12.A.1

The Pythagorean Theorem can be applied to right triangles with points on the unit circle at coordinates (cos⁡θ,sin⁡θ)\left(\cos \theta, \sin \theta\right), resulting in the Pythagorean identity: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.

Essential Knowledge: 3.12.A.2

The Pythagorean identity can be algebraically manipulated into other forms involving trigonometric functions, such as tan⁡2θ=sec⁡2θ−1\tan^2 \theta = \sec^2 \theta - 1, and can be used to establish other trigonometric relationships, such as arcsin⁡x=arccos⁡(1−x2)\arcsin x = \arccos\left(\sqrt{1 - x^2}\right), with appropriate domain restrictions.

Learning Objective: 3.12.B

Rewrite trigonometric expressions in equivalent forms with sine and cosine sum identities.

Essential Knowledge: 3.12.B.1

The sum identity for sine is sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta.

Essential Knowledge: 3.12.B.2

The sum identity for cosine is cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta.

Essential Knowledge: 3.12.B.3

The sum identities for sine and cosine can also be used as difference and double-angle identities.

Essential Knowledge: 3.12.B.4

Properties of trigonometric functions, known trigonometric identities, and other algebraic properties can be used to verify additional trigonometric identities.

Learning Objective: 3.12.C

Solve equations using equivalent analytic representations of trigonometric functions.

Essential Knowledge: 3.12.C.1

A specific equivalent form involving trigonometric expressions can make information more accessible.

Essential Knowledge: 3.12.C.2

Equivalent trigonometric forms may be useful in solving trigonometric equations and inequalities.

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