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 $\sin(2\theta)$ , $\cos(2\theta)$ , and $\tan(2\theta)$ in terms of single-angle trig functions.

What if both angles in the sum formula are the same? Setting $A = B = \theta$ 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

\cos(\theta) = \frac{3}{5}

, find

\cos(2\theta)

using the double-angle formula.

Example 2

hard

\sin\theta + \cos\theta = \tfrac{1}{2}

, find

\sin(2\theta)

Example 3

hard

Solve

\cos(2x) = \sin x

[0, 2\pi)

Example 4

easy

Fill in:

\sin(2\theta) =

___.

Example 5

medium

\cos\theta = \tfrac{7}{25}

and

\theta

is in Quadrant IV, find

\cos(2\theta)

Example 6

easy

Give the form of

\cos(2\theta)

in terms of

\sin^2\theta

only.

Example 7

medium

Simplify

\cos^2(x) - \sin^2(x)

using a double-angle identity.

Example 8

hard

Solve

\sin(2x) = \cos(x)

for

x \in [0, 2\pi)

Example 9

easy

\sin(2\theta) = 2\sin\theta

correct? Answer yes or no.

Example 10

medium

Express

\sin(4\theta)

using a double-angle formula once.

Example 11

medium

\tan\theta = \frac13

, find

\tan(2\theta)

Example 12

medium

\sin\theta = -\tfrac{3}{5}

and

\theta

is in Quadrant III, find

\sin(2\theta)

Example 13

medium

\tan\theta = -\tfrac{1}{2}

, find

\tan(2\theta)

Example 14

medium

\cos\theta = \tfrac{12}{13}

and

\theta

is in Quadrant I, find

\tan(2\theta)

Example 15

easy

State the double-angle formula for

\tan(2\theta)

Example 16

easy

\cos\theta = \frac35

, use a double-angle form to find

\cos(2\theta)

Example 17

easy

Fill in:

\cos(2\theta) = 2\cos^2\theta -

___.

Example 18

medium

Use power-reduction to write

\cos^2 x

in terms of

\cos(2x)

Example 19

medium

\sin\theta = \frac{5}{13}

and

\theta

is in Quadrant I, find

\cos(2\theta)

Example 20

medium

Simplify

\frac{2\tan 22.5°}{1 - \tan^2 22.5°}

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Related Concepts

Sum and Difference Identities