Practice Double-Angle Identities in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

Formulas expressing sin⁡(2θ)\sin(2\theta), cos⁡(2θ)\cos(2\theta), and tan⁡(2θ)\tan(2\theta) in terms of single-angle trig functions.

What if both angles in the sum formula are the same? Setting A=B=θA = B = \theta in the sum identities gives you the double-angle formulas. They answer: if you know the trig values for an angle, what are the trig values for twice that angle? The cosine double-angle formula is especially versatile because it has three equivalent forms, each useful in different situations—pick whichever one simplifies your problem.

Showing a random 20 of 50 problems.

Example 1

easy
If cos⁡(θ)=35\cos(\theta) = \frac{3}{5}, find cos⁡(2θ)\cos(2\theta) using the double-angle formula.

Example 2

hard
If sin⁡θ+cos⁡θ=12\sin\theta + \cos\theta = \tfrac{1}{2}, find sin⁡(2θ)\sin(2\theta).

Example 3

hard
Solve cos⁡(2x)=sin⁡x\cos(2x) = \sin x on [0,2π)[0, 2\pi).

Example 4

easy
Fill in: sin⁡(2θ)=\sin(2\theta) = ___.

Example 5

medium
If cos⁡θ=725\cos\theta = \tfrac{7}{25} and θ\theta is in Quadrant IV, find cos⁡(2θ)\cos(2\theta).

Example 6

easy
Give the form of cos⁡(2θ)\cos(2\theta) in terms of sin⁡2θ\sin^2\theta only.

Example 7

medium
Simplify cos⁡2(x)−sin⁡2(x)\cos^2(x) - \sin^2(x) using a double-angle identity.

Example 8

hard
Solve sin⁡(2x)=cos⁡(x)\sin(2x) = \cos(x) for x∈[0,2π)x \in [0, 2\pi).

Example 9

easy
Is sin⁡(2θ)=2sin⁡θ\sin(2\theta) = 2\sin\theta correct? Answer yes or no.

Example 10

medium
Express sin⁡(4θ)\sin(4\theta) using a double-angle formula once.

Example 11

medium
If tan⁡θ=13\tan\theta = \frac13, find tan⁡(2θ)\tan(2\theta).

Example 12

medium
If sin⁡θ=−35\sin\theta = -\tfrac{3}{5} and θ\theta is in Quadrant III, find sin⁡(2θ)\sin(2\theta).

Example 13

medium
If tan⁡θ=−12\tan\theta = -\tfrac{1}{2}, find tan⁡(2θ)\tan(2\theta).

Example 14

medium
If cos⁡θ=1213\cos\theta = \tfrac{12}{13} and θ\theta is in Quadrant I, find tan⁡(2θ)\tan(2\theta).

Example 15

easy
State the double-angle formula for tan⁡(2θ)\tan(2\theta).

Example 16

easy
If cos⁡θ=35\cos\theta = \frac35, use a double-angle form to find cos⁡(2θ)\cos(2\theta).

Example 17

easy
Fill in: cos⁡(2θ)=2cos⁡2θ−\cos(2\theta) = 2\cos^2\theta - ___.

Example 18

medium
Use power-reduction to write cos⁡2x\cos^2 x in terms of cos⁡(2x)\cos(2x).

Example 19

medium
If sin⁡θ=513\sin\theta = \frac{5}{13} and θ\theta is in Quadrant I, find cos⁡(2θ)\cos(2\theta).

Example 20

medium
Simplify 2tan⁡22.5°1−tan⁡222.5°\frac{2\tan 22.5°}{1 - \tan^2 22.5°}.
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