Double Angle Identities Calculator
Verify every double-angle identity at a specific angle: see sin(2θ), all three equivalent forms of cos(2θ), tan(2θ), and the half-angle values derived from them.
Angle unit
All three cos(2θ) identities evaluate to the same value
- 1
Convert to radians
60 × π ÷ 180 = 1.047198 - 2
cos θ
cos(1.0472) = 0.5 - 3
sin θ
sin(1.0472) = 0.866025 - 4
cos(2θ) = cos²θ − sin²θ
(0.5)² − (0.866025)² = -0.500000All three cos(2θ) forms evaluate to this same value.
How does this calculator work?
Double-angle identities: sin(2θ) = 2 sin θ cos θ; cos(2θ) takes three identical forms — cos²θ − sin²θ, 2cos²θ − 1, and 1 − 2sin²θ; tan(2θ) = 2 tan θ / (1 − tan²θ). Half-angle values follow by rearranging the cosine form. Enter an angle to verify all identities numerically.
Formula
How this is calculated
Double-angle identities connect a trigonometric function of 2θ to functions of the original angle θ. They are proved by applying the angle-addition formulas with A = B = θ: sin(2θ) = 2 sin θ cos θ (from the sine addition rule) and cos(2θ) = cos²θ − sin²θ (from the cosine addition rule). Substituting the Pythagorean identity sin²θ + cos²θ = 1 into the cosine form yields two more equivalent identities — 2cos²θ − 1 and 1 − 2sin²θ — all numerically identical.
This calculator evaluates every identity at the specified angle so you can confirm they agree. It also computes the half-angle values sin(θ/2) = ±√((1 − cos θ)/2) and cos(θ/2) = ±√((1 + cos θ)/2), which are themselves derived by rearranging the 2cos²(θ/2) − 1 and 1 − 2sin²(θ/2) forms of the cosine double-angle identity. The displayed half-angle results are the positive square roots; the correct sign depends on which quadrant θ/2 falls in.
Degrees are converted to radians before evaluation. tan(2θ) is undefined when tan²θ = 1 (θ = 45°, 135°, etc.), because the denominator 1 − tan²θ equals zero there. Any apparent disagreement among the three cos(2θ) forms beyond 15 decimal places is floating-point noise, not a mathematical error.
Frequently asked questions
All three forms are the same expression with the Pythagorean identity sin²θ + cos²θ = 1 applied differently. Starting from cos²θ − sin²θ, replace sin²θ with 1 − cos²θ to get 2cos²θ − 1, or replace cos²θ with 1 − sin²θ to get 1 − 2sin²θ.
Rewrite the double-angle identity cos(2α) = 1 − 2sin²α as sin²α = (1 − cos 2α)/2, then substitute α = θ/2 to get sin²(θ/2) = (1 − cos θ)/2. Taking the square root gives the half-angle formula for sine; the cosine form follows similarly.
The sign of sin(θ/2) and cos(θ/2) depends on which quadrant θ/2 falls in. The calculator always shows the positive root as a reference value; flip the sign if θ/2 is in a quadrant where the function is negative.
TG we-Calculate Editorial Team. (2026). Double Angle Identities Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/double-angle-identities-calculator
TG we-Calculate Editorial Team. "Double Angle Identities Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/double-angle-identities-calculator.
TG we-Calculate Editorial Team, "Double Angle Identities Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/double-angle-identities-calculator
@misc{wecalculate_double_angle_identities_calculator, title = {Double Angle Identities Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/double-angle-identities-calculator}}, year = {2026}, note = {TG we-Calculate} }
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