Precalculus Intermediate

Double & Half Angle Formulas Calculator

Compute sin(2θ), cos(2θ), tan(2θ) and sin(θ/2), cos(θ/2), tan(θ/2) from a given angle θ or from known values of sin θ and cos θ. Shows all three forms of cos(2θ), half-angle sign determination by quadrant, and full step-by-step substitution with exact values.

Live Calculator · Exact Values · All Three cos(2θ) Forms · Step-by-Step · Precalculus
Inputs
Common Examples
Enter sin θ and cos θ as decimals or fractions (e.g. 3/5, −4/5, 0.6). The Pythagorean identity sin²θ + cos²θ = 1 will be verified.
Common Values
Results
Enter an angle θ (or sin θ and cos θ), then press Calculate to see all double and half angle values.
Double Angle (2θ)
sin(2θ)
exact
—
cos(2θ)
exact
—
tan(2θ)
exact
—
Half Angle (θ/2)
sin(θ/2)
exact
—
cos(θ/2)
exact
—
tan(θ/2)
exact
—
Step-by-Step Derivation
Double Angle Formulas
sin(2θ) = 2 sin θ cos θ
tan(2θ) = 2 tan θ / (1 − tan²θ)
cos(2θ) — three equivalent forms = cos²θ − sin²θ
= 2 cos²θ − 1
= 1 − 2 sin²θ
These come from the sum formulas with A = B = θ. For example, sin(A+B) = sin A cos B + cos A sin B becomes sin(2θ) = 2 sin θ cos θ. The three cos(2θ) forms use the identity sin²θ + cos²θ = 1 to substitute.
Half Angle Formulas & the ± Sign Rule
sin(θ/2) = ±√((1 − cos θ)/2)
cos(θ/2) = ±√((1 + cos θ)/2)
tan(θ/2) — two forms (no ± needed) = sin θ / (1 + cos θ)
= (1 − cos θ) / sin θ
Sign Determination for θ/2
θ in range θ/2 in range sin(θ/2) cos(θ/2)
Q I (0°–90°) 0°–45° (Q I) + +
Q II (90°–180°) 45°–90° (Q I) + +
Q III (180°–270°) 90°–135° (Q II) + −
Q IV (270°–360°) 135°–180° (Q II) + −
For θ in [0°, 360°], θ/2 always falls in [0°, 180°], so sin(θ/2) is always positive. cos(θ/2) is positive when θ/2 is in Q I (θ < 180°) and negative when θ/2 is in Q II (θ ≥ 180°). The tan(θ/2) formulas avoid the ± entirely.
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