How Trig Identities Work
A trigonometric identity is an equation involving sin, cos, tan, and their reciprocals that is true for every value of the angle, not just for special cases. This calculator takes one angle θ and applies the fundamental identities — the quotient identity, the reciprocal identities, the Pythagorean identity, and the co-function identity — to compute all six trig functions and show that the identities hold numerically for your input.
Formula and method
Starting from sin θ and cos θ (the y and x coordinates of the point where the terminal side of angle θ meets the unit circle), the calculator derives everything else algebraically: the quotient identity gives tan θ = sin θ / cos θ; the reciprocal identities give csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ = cos θ / sin θ; the Pythagorean identity sin²θ + cos²θ = 1 comes directly from the unit circle's equation x² + y² = 1; and the co-function identity sin(90° − θ) = cos θ comes from the fact that the two non-right angles of a right triangle always sum to 90°, so each angle's sine equals its complement's cosine.
Common sources of error
- Degrees vs. radians: select the correct unit for your angle — evaluating a degree value as if it were radians (or vice versa) gives a completely different, wrong answer.
- Undefined values: tan θ and sec θ are undefined at θ = 90°, 270°, ... (where cos θ = 0); cot θ and csc θ are undefined at θ = 0°, 180°, 360°, ... (where sin θ = 0). These show as "undefined," not zero or an error.
- Sign errors when inverting the Pythagorean identity: solving sin²θ + cos²θ = 1 for cos θ gives cos θ = ±√(1 − sin²θ) — you must use the quadrant of θ to pick the correct sign, not just take the positive root.
Applications
Trig identities are the algebra that lets you simplify expressions, solve trigonometric equations, and rewrite one function in terms of another without a calculator — for example, proving that (1 − cos²θ)/sin θ = sin θ, or converting a tan-based formula into an equivalent sin/cos form. They also underlie physics (wave and oscillation equations), engineering (AC circuit phase analysis), and navigation and surveying calculations that combine angles from multiple triangles.