Trig Identities Calculator
Trigonometric identities are equations that hold for every value of the angle, and they are the backbone of simplifying expressions, solving equations and proving results across trigonometry, calculus and physics. This calculator lets you enter an angle in degrees and then evaluates the core identities at that angle so you can see them hold numerically. It computes the sine, cosine and tangent of the angle, then checks the Pythagorean identity, sine squared plus cosine squared equals one, and the double-angle identity, sine of twice the angle equals two times sine times cosine. Seeing the two sides of an identity produce the same number is a quick, concrete way to confirm the identity and to build intuition for why it is always true. Because trigonometric functions repeat and can be negative, the signs of the individual values change from quadrant to quadrant, yet the identities still balance, which is exactly what makes them identities rather than ordinary equations. Every value here is computed deterministically from the angle you enter using the standard trigonometric functions, never estimated or guessed, so the same angle always returns the same numbers. The identities used and a worked example at the default angle that reconciles to the calculator are shown in full below.
The Pythagorean identity states sin^2(t) + cos^2(t) = 1 for every angle t. At 30 degrees, sin is 0.50 and cos is 0.87, and 0.50^2 + 0.87^2 rounds to 1.00, confirming the identity.
Formula
Pythagorean: sin^2(t) + cos^2(t) = 1
Double angle: sin(2t) = 2 sin(t) cos(t)
t is the angle, converted to radians as t x pi / 180
The angle is converted from degrees to radians, then the standard sine and cosine are evaluated. The Pythagorean identity sums the squares (always 1), and the double-angle identity compares sin(2t) with 2 sin(t) cos(t).
Worked example
Evaluate the identities at an angle of 30 degrees.
- sin(30 deg) = 0.5000 and cos(30 deg) = 0.8660
- sin^2 + cos^2 = 0.25 + 0.75 = 1.00, confirming the Pythagorean identity
- sin(60 deg) = 0.8660, and 2 x 0.5 x 0.8660 = 0.8660, confirming the double-angle identity
This is the calculator's default angle, so the values above match the widget exactly.
Trig Identities Calculator: frequently asked questions
What is the Pythagorean identity?
The Pythagorean identity states that for any angle, the square of its sine plus the square of its cosine equals one. It follows directly from the definition of sine and cosine on the unit circle.
What is the double-angle formula for sine?
The sine of twice an angle equals two times the sine of the angle times the cosine of the angle. This calculator verifies it numerically at any angle you enter.
Why do the signs change with the angle?
Sine and cosine are positive or negative depending on the quadrant of the angle, but the identities still balance because both sides change consistently. That is why they hold for every angle.
Does the calculator use degrees or radians?
You enter the angle in degrees. The calculator converts it to radians internally, since the standard trigonometric functions operate in radians.
Are the values exact?
They are computed to the decimal places shown using standard double-precision trigonometry. Tiny rounding at the last digit can occur but does not affect the displayed values.
Official sources
- Trigonometric function and identity reference: US National Institute of Standards and Technology (NIST). As at 25 June 2026.
Reviewed by the CalculatorHub team, edited by James Graham, 25 June 2026. See our methodology. This is general information, not financial, tax, legal or investment advice.