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Trigonometry

Double-angle formulas

The sine and cosine of twice an angle, written using the sine and cosine of the angle itself.

sin⁡2θ=2sin⁡θcos⁡θ, cos⁡2θ=cos2⁡θ−sin2⁡θ
Cosine, using only sinecos⁡2θ=1−2sin2⁡θ
Cosine, using only cosinecos⁡2θ=2cos2⁡θ−1
Tangenttan⁡2θ=2tan⁡θ1−tan2⁡θ

Calculator

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−1.5−1−0.50.511.5−1−0.50.51θ2θ
  • sin 2θ = 2 sin θ cos θ = 2 × 0.5 × 0.866 = 0.866
  • cos 2θ = cos²θ − sin²θ = 0.5
  • Doubling the angle does not double the sine: sin θ = 0.5 but sin 2θ = 0.866

What the symbols mean

θ
the angle
sin⁡(2θ)
the sine of the doubled angle
cos⁡(2θ)
the cosine of the doubled angle

When to use it

Use them to simplify expressions with sin⁡2x or cos⁡2x, to solve equations such as sin⁡2x=sin⁡x (rewrite, then factorise), to integrate sin2⁡x and cos2⁡x (rearrange cos⁡2x=1−2sin2⁡x), and in physics, where the range of a projectile depends on sin⁡2θ.

Worked example

If θ = 30°, find sin 2θ using sin 2θ = 2 sin θ cos θ.

  1. Use the double-angle formula for sine.

    sin⁡(2θ)=2sin⁡θcos⁡θ

  2. Substitute θ and work it out.

    sin⁡(2θ)=2⁢sin⁡(θ)⁢cos⁡(θ)
    =2⁢sin⁡(30°)⁢cos⁡(30°)
    =2·0.5⁢cos⁡(30°)
    =1⁢cos⁡(30°)
    ≈1·0.866025
    ≈0.866025

  3. Check directly: the angle 2θ is 60°, and its sine is 0.866025.

Answer The sine of the doubled angle ≈ 0.8660

Why it works

They are the angle-sum formulas with both angles equal: sin⁡(θ+θ)=sin⁡θcos⁡θ+cos⁡θsin⁡θ. The other two cosine forms come from replacing cos2⁡θ or sin2⁡θ with the Pythagorean identity.

Common mistakes

  • Doubling the sine: sin⁡2θ is not 2sin⁡θ (try θ = 90°: 0 against 2).
  • Mixing up the cosine forms: it is 1−2sin2⁡θ, not 1−sin2⁡θ (that is just cos2⁡θ).
  • Losing solutions when solving: sin⁡2θ=k has two solutions for 2θ in each turn, so twice as many for θ.