Double-angle formulas
cos 2θ = cos² θ − sin² θ = 2cos² θ − 1 = 1 − 2sin² θ
tan 2θ = 2 tan θ / (1 − tan² θ)
They come from the sum formulas with both angles equal: sin(θ + θ) = sin θ cos θ + cos θ sin θ = 2 sin θ cos θ, and cos(θ + θ) = cos² θ − sin² θ. The other two forms of cos 2θ use sin² θ + cos² θ = 1.
Half-angle formulas
cos(θ/2) = ±√((1 + cos θ)/2)
tan(θ/2) = (1 − cos θ)/sin θ = sin θ/(1 + cos θ)
The sign depends on the quadrant of θ/2, not θ. The tan formulas need no ± sign.
Power-reducing formulas
These are the cos 2θ formulas solved for sin² θ and cos² θ. They're the standard way to integrate sin² x: ∫sin² x dx = ∫(1 − cos 2x)/2 dx = x/2 − sin(2x)/4 + C.
Worked examples
1. Exact value of sin 15°. 15° is half of 30°, and it's in the first quadrant, so use the + sign: sin 15° = √((1 − cos 30°)/2) = √((1 − √3/2)/2) = √(2 − √3)/2. This also equals (√6 − √2)/4 ≈ 0.258819.
2. Exact value of cos 22.5°. cos 22.5° = √((1 + cos 45°)/2) = √((1 + √2/2)/2) = √(2 + √2)/2 ≈ 0.92388.
3. tan(π/8). tan(π/8) = (1 − cos(π/4))/sin(π/4) = (1 − √2/2)/(√2/2) = √2 − 1 ≈ 0.414214.
4. Using a known sine. If sin θ = 3/5 and θ is in quadrant II, then cos θ = −4/5. So:
- sin 2θ = 2(3/5)(−4/5) = −24/25
- cos 2θ = (−4/5)² − (3/5)² = 16/25 − 9/25 = 7/25
- tan 2θ = sin 2θ / cos 2θ = −24/7
Check: (−24/25)² + (7/25)² = 576/625 + 49/625 = 1, as it should.
5. Simplifying. (sin 2x)/(1 + cos 2x) = (2 sin x cos x)/(2cos² x) = tan x.
Solving equations with double angles
Replacing sin 2x or cos 2x with a single-angle form usually lets you factor. Solve sin 2x = sin x for 0 ≤ x < 2π:
- Rewrite: 2 sin x cos x = sin x.
- Move everything to one side and factor. Don't divide by sin x, or you'll lose solutions: sin x (2 cos x − 1) = 0.
- sin x = 0 gives x = 0 or π. cos x = 1/2 gives x = π/3 or 5π/3.
- So x = 0, π/3, π, 5π/3.
For cos 2x + cos x = 0, choose the form of cos 2x with only cosines: 2cos² x − 1 + cos x = 0 factors as (2 cos x − 1)(cos x + 1) = 0, giving x = π/3, π and 5π/3.
Quick reference
| Angle θ | sin 2θ | cos 2θ | sin(θ/2) | cos(θ/2) |
|---|---|---|---|---|
| 30° | √3/2 | 1/2 | (√6 − √2)/4 | (√6 + √2)/4 |
| 45° | 1 | 0 | √(2 − √2)/2 | √(2 + √2)/2 |
| 60° | √3/2 | −1/2 | 1/2 | √3/2 |
| 90° | 0 | −1 | √2/2 | √2/2 |
Common mistakes
- sin 2θ is not 2 sin θ. For θ = 30°, sin 60° ≈ 0.866 but 2 sin 30° = 1.
- Choosing the half-angle sign from θ instead of θ/2. If θ = 300°, θ/2 = 150° is in quadrant II, so sin(θ/2) is positive and cos(θ/2) is negative.
- Mixing degrees and radians on a calculator. See degrees and radians.
Read exact values off the unit circle, evaluate trig functions with the scientific calculator, and find angles from values with the inverse trig calculator. To differentiate the results, see the derivative rules table.
Sources
- OpenStax Precalculus 2e, 7.3: Double-angle, half-angle and reduction formulas
- OpenStax Precalculus 2e, 7.2: Sum and difference identities
Every worked example on this page was checked independently with SymPy.