Double-Angle Identities Math Example 2

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Example 2

medium
Find sin⁡(2θ)\sin(2\theta) given that tan⁡(θ)=512\tan(\theta) = \frac{5}{12} and θ\theta is in Quadrant I.

Solution

  1. 1
    The double-angle formula for sine is sin⁡(2θ)=2sin⁡(θ)cos⁡(θ)\sin(2\theta) = 2\sin(\theta)\cos(\theta).
  2. 2
    From tan⁡θ=512\tan\theta = \frac{5}{12} in QI, the reference triangle has opposite =5= 5, adjacent =12= 12, hypotenuse =25+144=13= \sqrt{25+144} = 13.
  3. 3
    So sin⁡θ=513\sin\theta = \frac{5}{13} and cos⁡θ=1213\cos\theta = \frac{12}{13}.
  4. 4
    sin⁡(2θ)=2⋅513⋅1213=120169\sin(2\theta) = 2 \cdot \frac{5}{13} \cdot \frac{12}{13} = \frac{120}{169}.

Answer

sin⁡(2θ)=120169\sin(2\theta) = \frac{120}{169}
To use the sine double-angle formula, we need both sin⁡θ\sin\theta and cos⁡θ\cos\theta. Given tangent, we can reconstruct the right triangle to find these values. This is a common multi-step process in trigonometry problems.

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 1 easy

If [formula], find [formula] using the double-angle formula.

Example 3 medium

Simplify [formula] using a double-angle identity.

Example 4 hard

Solve [formula] for [formula].

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