Trigonometric Identities Calculator

Trigonometric identities connect the six functions used to describe an angle on the unit circle. A point on that circle has coordinates (cos θ, sin θ), so cosine is the horizontal coordinate and sine is the vertical coordinate. Tangent is their quotient, sin θ / cos θ. The three reciprocal functions follow directly: cosecant is 1 / sin θ, secant is 1 / cos θ, and cotangent is cos θ / sin θ. Knowing any one valid function value plus the quadrant is enough to recover all the others. The central relationship is the Pythagorean identity sin² θ + cos² θ = 1. If sine is known, cosine's magnitude is the square root of 1 - sin² θ. If cosine is known, sine is found the same way. A known tangent determines a proportional point and normalization places it on the unit circle. Reciprocal inputs are first inverted, after which the same rules apply. These steps preserve the mathematical relationships rather than estimating an angle and converting it back. Quadrants determine signs. In quadrant I, sine and cosine are both positive. In quadrant II, sine is positive and cosine is negative. In quadrant III, both are negative, and in quadrant IV, sine is negative while cosine is positive. Tangent and cotangent are positive when sine and cosine have matching signs. The calculator checks the sign of the entered value against the selected quadrant so an impossible combination is not presented as a valid answer. Some functions are undefined at axis angles. For example, tangent and secant are undefined where cosine equals zero, while cotangent and cosecant are undefined where sine equals zero. Very large reciprocal results are displayed as undefined because floating-point approximations near zero should not be mistaken for meaningful finite values. This identity calculator is useful for precalculus exercises, unit-circle practice, physics components, engineering formulas, and checking hand calculations. It reports decimal values to six places, which is convenient for practical work. For symbolic proofs, retain exact radicals and fractions when possible, and use this numerical result as a verification rather than a replacement for algebraic reasoning.