Mathematics.

trigonometric identities

Double-Angle Formulas

Trigonometry20 minDifficulty3 out of 10

You should know: trigonometric identities

Overview

The double-angle formulas express sin(2θ), cos(2θ), and tan(2θ) in terms of sin θ, cos θ, and tan θ. They arise as the special case A = B of the sum formulas and appear throughout calculus (integrating powers of sine and cosine), physics (wave intensity, which depends on amplitude squared), and engineering (power calculations in AC circuits). Cosine's double-angle formula has three equivalent forms, related by the Pythagorean identity, each useful in different contexts.

Intuition

Doubling an angle is just adding it to itself, so plugging A = B into the sum-of-angles formulas directly produces the double-angle identities — there is no new geometry here, only algebraic simplification. The three forms of cos(2θ) are all the same fact viewed through the Pythagorean identity: since sin²θ+cos²θ=1, any one of the forms can be rewritten as either of the others, which is why you can choose whichever form eliminates the variable you don't want (e.g., using 2cos²θ−1 to solve for cosθ alone).

Formal Definition

Definition

Setting A = B = θ in the sum formulas gives:

sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta
Double-angle sine
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
Double-angle cosine (three forms)
tan⁡(2θ)=2tan⁡θ1−tan⁡2θ\tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta}
Double-angle tangent

Worked Examples

  1. Apply the double-angle sine formula.

    sin⁡(2θ)=2⋅35⋅45=2425\sin(2\theta) = 2\cdot\frac{3}{5}\cdot\frac{4}{5} = \frac{24}{25}

  2. Apply the double-angle cosine formula (difference-of-squares form).

    cos⁡(2θ)=(45)2−(35)2=1625−925=725\cos(2\theta) = \left(\frac{4}{5}\right)^2 - \left(\frac{3}{5}\right)^2 = \frac{16}{25}-\frac{9}{25} = \frac{7}{25}

Answer: sin(2θ) = 24/25, cos(2θ) = 7/25

Practice Problems

Difficulty 3/10

Given sinθ = 3/5 and cosθ = 4/5, find cos(2θ) using the form 1 − 2sin²θ, and verify it matches 2cos²θ − 1.

Difficulty 4/10

Find the exact value of cos(120°) using cos(2·60°) and cos(60°) = 1/2.

Difficulty 6/10

The intensity of a light wave through a polarizer is proportional to cos²θ (Malus's law). Use the double-angle identity to rewrite cos²θ and find the intensity fraction when θ = 30°.

Quiz

Which is NOT an equivalent form of cos(2θ)?
If sinθ = 3/5 and cosθ = 4/5, then sin(2θ) equals:
The double-angle formulas are derived by:

Summary

  • sin(2θ) = 2sinθcosθ, and cos(2θ) has three equivalent forms: cos²θ−sin²θ, 2cos²θ−1, 1−2sin²θ.
  • They follow directly from the sum formulas by setting the two angles equal.
  • They're essential for calculus (power-reduction), physics (intensity ∝ amplitude²), and AC power calculations.

References

  1. WebsiteWikipedia — List of trigonometric identities