Mathematics.

trigonometric identities

Half-Angle Formulas

Trigonometry25 minDifficulty4 out of 10

You should know: double angle formulas

Overview

The half-angle formulas express sin(θ/2), cos(θ/2), and tan(θ/2) in terms of cos θ. They are obtained by solving the double-angle cosine formula for sin²θ and cos²θ and then substituting θ/2 for θ, which is why they involve a square root and require choosing the correct sign based on which quadrant θ/2 falls in. They're used to find exact values at angles like 15° or 22.5° and to simplify integrals involving trigonometric functions via the Weierstrass (tangent half-angle) substitution.

Intuition

The double-angle formula cos(2α) = 1 − 2sin²α is really a statement about any angle and its double. Relabeling 2α as θ (so α = θ/2) and solving for sinα gives the half-angle sine formula directly. The square root — and the resulting ± ambiguity — appears because squaring loses sign information: cos(2α) only tells you sin²α, not sinα itself, so you must independently determine which sign is correct by checking which quadrant θ/2 lies in.

Formal Definition

Definition

For any angle θ, with the sign chosen according to the quadrant of θ/2:

sin⁡(θ2)=±1−cos⁡θ2\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1-\cos\theta}{2}}
Half-angle sine
cos⁡(θ2)=±1+cos⁡θ2\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1+\cos\theta}{2}}
Half-angle cosine
tan⁡(θ2)=1−cos⁡θsin⁡θ=sin⁡θ1+cos⁡θ\tan\left(\frac{\theta}{2}\right) = \frac{1-\cos\theta}{\sin\theta} = \frac{\sin\theta}{1+\cos\theta}
Half-angle tangent (sign-free forms)

Worked Examples

  1. θ/2 = 15° is in quadrant I, so cosine is positive; use cos30° = √3/2.

    cos⁡(15°)=1+cos⁡30°2=1+322\cos(15°) = \sqrt{\frac{1+\cos30°}{2}} = \sqrt{\frac{1+\frac{\sqrt3}{2}}{2}}

  2. Simplify the nested fraction.

    =2+34=2+32= \sqrt{\frac{2+\sqrt3}{4}} = \frac{\sqrt{2+\sqrt3}}{2}

Answer: cos(15°) = √(2+√3)/2 ≈ 0.9659

Practice Problems

Difficulty 4/10

Find the exact value of sin(15°) using the half-angle formula with θ = 30°.

Difficulty 5/10

Given cosθ = 7/25 with θ in quadrant I (so θ/2 is also in quadrant I), find cos(θ/2).

Difficulty 6/10

A ramp's incline angle θ satisfies cosθ = 0.8. Using the half-angle formula, find tan(θ/2), which is used to compute the grade of a switchback that halves the incline.

Quiz

The half-angle formula for cos(θ/2) requires a ± sign because:
cos(15°), found via the half-angle formula from θ = 30°, equals:
The sign-free half-angle tangent formula is:

Summary

  • Half-angle formulas solve the double-angle cosine identity for sin(θ/2) and cos(θ/2), introducing a ± resolved by the quadrant of θ/2.
  • The tangent half-angle formula tan(θ/2) = (1−cosθ)/sinθ avoids sign ambiguity entirely.
  • They give exact values at angles like 15° and underlie the Weierstrass substitution used in calculus.

References

  1. WebsiteWikipedia — List of trigonometric identities