Double Angle Formula

A double-angle formula is an exact identity derived from sum formulas. It preserves value while changing the angle argument from 2θ to expressions in θ. Nothing about the angle size changes; only the way you write the trigonometric value changes.

Identity relationships link sine, cosine, and tangent forms. The three cosine versions are algebraically equivalent because cos² θ + sin² θ = 1. You can move between them by substituting the Pythagorean identity when needed.

Knowing all three cosine versions helps you pick the fastest route when only sin θ or only cos θ is available. In a triangle problem where cos θ = 3/5 is given, cos(2θ) = 2 cos² θ − 1 avoids finding sin θ first. In a proof that already contains sin² θ and cos² θ, the difference form cos(2θ) = cos² θ − sin² θ may be the cleanest step.

Tangent’s double-angle formula comes from dividing the sine identity by the cosine identity, with care taken when cos(2θ) = 0. That restriction matters in calculus when you rewrite tan(2θ) and need to exclude angles where the denominator vanishes.

Sine double-angle formula:

sin(2θ) = 2 sin θ cos θ

Cosine double-angle formulas:

• cos(2θ) = cos² θ − sin² θ
• cos(2θ) = 2 cos² θ − 1
• cos(2θ) = 1 − 2 sin² θ

Tangent double-angle formula:

tan(2θ) = 2 tan θ / (1 − tan² θ)

Each cosine form is useful in a different context. The first highlights symmetry between squared terms. The second is ideal when cos θ alone is known. The third appears often when a problem gives sin θ or sin² θ.

These are the same identities discussed from a transformation angle in Double Angle Identities, where the focus is rewriting expressions rather than listing formulas.

1. Write the target function. State clearly whether you need sin(2θ), cos(2θ), or tan(2θ).
2. List what you know about θ. Gather sin θ, cos θ, tan θ, or squared versions from the unit circle, a triangle, or the problem statement.
3. Choose the identity form with the fewest unknown ratios. Pick the cosine version that matches your given information to minimize extra steps.
4. Substitute and simplify. Plug in exact values when possible. Keep radicals in simplified form unless the problem asks for decimals.
5. Confirm with examples or the calculator. Compare against worked examples or the homepage tool for a numeric check.

Example 1: cosine: Find cos(2θ) when θ = 60° and cos 60° = 1/2.

Use cos(2θ) = 2 cos² θ − 1 because cosine at θ is given:

cos(120°) = 2(1/2)² − 1 = 2(1/4) − 1 = −1/2.

Direct evaluation confirms cos(120°) = −1/2.

Example 2: sine: θ = π/4 rad. Find sin(π/2).

sin(π/2) = 2 sin(π/4) cos(π/4) = 2(√2/2)(√2/2) = 1.

Example 3: tangent: If tan θ = 1 and θ = 45°, then tan(90°) is undefined. The formula’s denominator 1 − tan² θ = 0, which correctly signals that tan(2θ) does not exist at this angle.