Double-Angle Identities Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Double-Angle Identities.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Double-angle formulas are special cases of the sum formulas with A = B. The cosine version has three equivalent forms, giving flexibility to match what information you have.

Common stuck point: Students often forget there are three forms of \cos(2\theta). The choice of form depends on context—pick the one that eliminates the trig function you don't need.

Sense of Study hint: If you only know sin, use cos(2x) = 1 - 2sin^2(x). If you only know cos, use cos(2x) = 2cos^2(x) - 1. Pick the form that matches what you have.

Worked Examples

Example 1

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

Solution

  1. 1
    Use the double-angle formula: \cos(2\theta) = 2\cos^2(\theta) - 1.
  2. 2
    Substitute: \cos(2\theta) = 2\left(\frac{3}{5}\right)^2 - 1 = 2 \cdot \frac{9}{25} - 1.
  3. 3
    = \frac{18}{25} - 1 = \frac{18}{25} - \frac{25}{25} = -\frac{7}{25}.

Answer

\cos(2\theta) = -\frac{7}{25}
The double-angle formula for cosine has three equivalent forms: \cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta. Choose the form that uses the information you have — here we used 2\cos^2\theta - 1 since we knew cosine.

Example 2

medium
Find \sin(2\theta) given that \tan(\theta) = \frac{5}{12} and \theta is in Quadrant I.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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

Example 2

hard
Solve \sin(2x) = \cos(x) for x \in [0, 2\pi).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

trig identities sum difference