Double-Angle Identities Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

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

Solution

  1. 1
    Use the double-angle formula: cos⁡(2θ)=2cos⁡2(θ)−1\cos(2\theta) = 2\cos^2(\theta) - 1.
  2. 2
    Substitute: cos⁡(2θ)=2(35)2−1=2⋅925−1\cos(2\theta) = 2\left(\frac{3}{5}\right)^2 - 1 = 2 \cdot \frac{9}{25} - 1.
  3. 3
    =1825−1=1825−2525=−725= \frac{18}{25} - 1 = \frac{18}{25} - \frac{25}{25} = -\frac{7}{25}.

Answer

cos⁡(2θ)=−725\cos(2\theta) = -\frac{7}{25}
The double-angle formula for cosine has three equivalent forms: cos⁡(2θ)=cos⁡2θ−sin⁡2θ=2cos⁡2θ−1=1−2sin⁡2θ\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 2cos⁡2θ−12\cos^2\theta - 1 since we knew cosine.

About Double-Angle Identities

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.

Learn more about Double-Angle Identities →

More Double-Angle Identities Examples

Example 2 medium

Find [formula] given that [formula] and [formula] is in Quadrant I.

Example 3 medium

Simplify [formula] using a double-angle identity.

Example 4 hard

Solve [formula] for [formula].

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