Trigonometry Calculator

About the Trigonometry Calculator

Type an angle in degrees, radians (pi/6, 5π/12) or gradians and get all six trigonometric functions at once. At the standard angles — every multiple of 15°, 18° and 22.5° — the values are exact: sin 75° = (√6 + √2)/4, cos 36° = (1 + √5)/4, tan 22.5° = √2 − 1. You also get the quadrant and the sign of each function, the reference angle, coterminal angles and the angle in every unit.

The inverse tab gives the principal value of arcsin, arccos, arctan, arccsc, arcsec and arccot — exact when the value is a standard one, so arcsin(√3/2) = 60° — and every angle in one turn with that value. The equation tab solves sin x = k and its relatives, including sin 2x = k, cos(x − 30°) = k and sin² x = k, with the principal value, the general solution for every integer n, and all the solutions in an interval you choose. The identity tab tests an identity such as sin 2x = 2 sin x cos x numerically at dozens of points and shows a counterexample when it fails.

Common uses

• Homework and exam revision: exact values, quadrant signs, reference angles and general solutions.
• Checking a step in a proof by testing an identity before trying to prove it.
• Engineering and physics: phase angles, components of vectors and inverse functions in degrees or radians.
• Surveying and navigation sums that mix degrees–minutes–seconds with decimals.

Exact values at standard angles

Exact values come from the 30-60-90 and 45-45-90 triangles, the half-angle and sum formulas, and the regular pentagon: sin 15° = (√6 − √2)/4, sin 18° = (√5 − 1)/4, sin 22.5° = √(2 − √2)/2, sin 30° = 1/2, sin 36° = √(10 − 2√5)/4 and sin 45° = √2/2. Every angle whose reference angle is 0°, 15°, 18°, 22.5°, 30°, 36°, 45°, 54°, 60°, 67.5°, 72°, 75° or 90° gets exact values for all six functions. Other angles get decimals, to the number of places you choose.

Quadrants, reference and coterminal angles

Angles are measured anticlockwise from the positive x-axis. In quadrant I all six functions are positive; in II only sin and csc; in III only tan and cot; in IV only cos and sec (“All Students Take Calculus”). The reference angle is the acute angle between the terminal side and the x-axis: θ, 180° − θ, θ − 180° or 360° − θ in quadrants I to IV. Coterminal angles differ by whole turns, θ ± n·360°, and have the same values.

Principal values and general solutions

Inverse functions return principal values as in the NIST Digital Library of Mathematical Functions, §4.23: arcsin and arccsc between −90° and 90°, arccos and arcsec between 0° and 180°, and arctan strictly between −90° and 90°. For arccot, most textbooks use 0° < θ < 180°, while DLMF defines arccot x = arctan(1/x) — you can choose either. From the principal value α, the general solutions are: sin x = sin α gives x = n·180° + (−1)ⁿα; cos x = cos α gives x = n·360° ± α; tan x = tan α gives x = n·180° + α; and sin² x = sin² α (likewise cos² and tan²) gives x = n·180° ± α, for every integer n.

Limitations

• Real angles and values only — no complex numbers.
• Exact forms are given at the standard angles; other angles, such as 10° or 1 radian, get decimal values in double precision (about 15 significant digits).
• The identity check is numerical: it compares both sides at dozens of points to 9 significant digits. That is strong evidence, not a proof, and it cannot see a difference that occurs only at isolated points.
• Equations of the form f(ax + b) = k are solved; sums of different functions, such as sin x + cos x = 1, are not.
• In degrees, π is just the number 3.14159…, so sin(π/2) is the sine of 1.5708°, not of 90°. Choose radians when π is part of an angle; the identity check points this out when it sees π in degree mode.

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