Trig Identities Calculator — Verify 7 Identities at Any Angle
Enter any angle in degrees or radians to see seven key trigonometric identities numerically verified at that angle — including the Pythagorean, quotient, reciprocal, double-angle (sin and cos) and half-angle (sin and cos) identities — with each formula and its computed values shown step by step.
Angle unit
Pythagorean identity — always equals exactly 1
Pythagorean identity: sin²θ + cos²θ = 1
Quotient identity: tan θ = sin θ / cos θ
Reciprocal identities: csc = 1/sin, sec = 1/cos, cot = cos/sin
Double-angle: sin(2θ) = 2 sin θ cos θ
Double-angle: cos(2θ) = cos²θ − sin²θ
Half-angle: sin(θ/2) = ±√((1 − cos θ)/2)
Half-angle: cos(θ/2) = ±√((1 + cos θ)/2)
- 1
sin(θ)
sin(0.523599 rad) = 0.5 - 2
cos(θ)
cos(0.523599 rad) = 0.866025 - 3
sin²θ + cos²θ
(0.5)² + (0.866025)² = 1Pythagorean identity — always equals exactly 1.
How does this calculator work?
Seven identities verified at your angle: Pythagorean (sin²+cos²=1), quotient (tan=sin/cos), three reciprocals (csc, sec, cot), two double-angle formulas (sin(2θ)=2sinθcosθ; cos(2θ)=cos²θ−sin²θ), and two half-angle formulas. Enter any angle in degrees or radians; each identity is shown with its numerical check.
Formula
How this is calculated
Trigonometric identities are equations that hold for all values of the angle. This calculator evaluates both sides of each identity numerically at your chosen angle, confirming they agree to machine precision. The identities verified are:
**Pythagorean**: sin²θ + cos²θ = 1 (derived from the unit circle definition). **Quotient**: tan θ = sin θ / cos θ (undefined at multiples of 90° where cos = 0). **Reciprocal**: csc = 1/sin, sec = 1/cos, cot = cos/sin. **Double-angle** (two forms): sin(2θ) = 2 sin θ cos θ, and cos(2θ) = cos²θ − sin²θ. **Half-angle** (magnitude only): |sin(θ/2)| = √((1−cos θ)/2) and |cos(θ/2)| = √((1+cos θ)/2); the sign depends on the quadrant of θ/2.
Numerical verification is not a mathematical proof — identities hold algebraically for all real angles (except where they are undefined), but showing them at one angle is a sanity check. To prove an identity generally, algebraic manipulation is required. The half-angle results show the absolute value; the actual sign is + or − depending on which quadrant θ/2 falls in.
Frequently asked questions
The Pythagorean identity sin²θ + cos²θ = 1 follows directly from the unit circle definition of sine and cosine: a point on the unit circle at angle θ has coordinates (cos θ, sin θ), and the circle equation x² + y² = 1 gives the identity immediately.
sin(2θ) = 2 sin θ cos θ. It is derived from the angle-addition formula sin(A+B) = sin A cos B + cos A sin B by setting A = B = θ. The double-angle formula for cos has three equivalent forms; the most common is cos(2θ) = cos²θ − sin²θ.
The ± reflects that the sign of sin(θ/2) or cos(θ/2) depends on the quadrant that θ/2 falls in, which cannot be determined from cos θ alone. If θ/2 is in Q1 or Q2 the sine is positive; if in Q3 or Q4 it is negative. This calculator shows the absolute value (always positive) for clarity.
Also known as
TG we-Calculate Editorial Team. (2026). Trig Identities Calculator — Verify 7 Identities at Any Angle [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/trig-identities-calculator
TG we-Calculate Editorial Team. "Trig Identities Calculator — Verify 7 Identities at Any Angle." TG we-Calculate. 2026. https://we-calculate.com/calculator/trig-identities-calculator.
TG we-Calculate Editorial Team, "Trig Identities Calculator — Verify 7 Identities at Any Angle," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/trig-identities-calculator
@misc{wecalculate_trig_identities_calculator, title = {Trig Identities Calculator — Verify 7 Identities at Any Angle}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/trig-identities-calculator}}, year = {2026}, note = {TG we-Calculate} }
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