Trigonometry Calculator
Find sine, cosine, tangent, cosecant, secant and cotangent of any angle in degrees, radians or gradians — with exact radical values for special angles, the unit circle, inverse and hyperbolic functions, and a right-triangle solver.
Written by Suhaib Hassan. Exact values and identities follow standard references and are checked by hand. Our methodology.
It converts the angle to radians, evaluates the function in double-precision floating point, and matches special angles to their exact radical forms.
It shows where a value comes from — the unit circle, the reference angle and the quadrant — not just the number.
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Unit circle
The point (cos θ, sin θ) on the circle of radius 1.
Step-by-step working
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Exact values of the special angles
Memorise these and you can derive most others by symmetry.
| θ (deg) | θ (rad) | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | undef. | 1 | undef. |
| 30° | π/6 | 1/2 | √3/2 | √3/3 | 2 | 2√3/3 | √3 |
| 45° | π/4 | √2/2 | √2/2 | 1 | √2 | √2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 | 2√3/3 | 2 | √3/3 |
| 90° | π/2 | 1 | 0 | undef. | 1 | undef. | 0 |
| 180° | π | 0 | −1 | 0 | undef. | −1 | undef. |
| 270° | 3π/2 | −1 | 0 | undef. | −1 | undef. | 0 |
Key identities
Pythagorean
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ
Follow from x² + y² = 1 on the unit circle.
Double angle
cos 2θ = cos²θ − sin²θ
tan 2θ = 2 tan θ ÷ (1 − tan²θ)
Euler’s formula
cosh x = (eˣ + e⁻ˣ) ÷ 2
sinh x = (eˣ − e⁻ˣ) ÷ 2
Links circular and hyperbolic functions.
Worked examples
Height of a tower
From 50 m away, the angle of elevation to the top is 35°.
Ladder
A 5 m ladder leans at 70° to the ground. How high does it reach?
Wave phase
At phase 210°, a sine wave of amplitude 1 has what value?
sin 210° = −1/2
What are the trigonometric functions?
For an angle θ in a right triangle, sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse and tan θ = opposite ÷ adjacent. Their reciprocals are csc θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ. On the unit circle, cos θ and sin θ are the x and y coordinates of the point at angle θ, which extends the definitions to all angles.
Degrees, radians and gradians
A full turn is 360° = 2π rad = 400 grad. Convert with rad = deg × π ÷ 180 and grad = deg × 10 ÷ 9. Calculus and most science use radians because derivatives such as d/dx sin x = cos x hold only in radians.
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
Sources
- NIST Digital Library of Mathematical Functions, chapter 4 — sections 4.14 (trigonometric functions) and 4.28 (hyperbolic functions).
- Euler, L. (1748). Introductio in analysin infinitorum, vol. 1, ch. 8 — trigonometric quantities arising from the circle.
- ISO 80000-2:2019, Quantities and units — Part 2: Mathematics.
Frequently asked questions
Why use radians instead of degrees in calculus?
Radian measure is the ratio of arc length to radius, so it needs no arbitrary scale factor. The key limit lim sin x ÷ x = 1 holds only in radians, and therefore d/dx sin x = cos x. In degrees an extra factor of π/180 appears in every derivative.
Why is tan 90° undefined?
tan θ = sin θ ÷ cos θ, and cos 90° = 0, so the quotient divides by zero. The tangent function has vertical asymptotes at odd multiples of 90°, where secant is also undefined.
What are the ranges of the inverse functions?
The principal values are: arcsin in −90° to 90°, arccos in 0° to 180°, arctan in −90° to 90° (exclusive), arccot in 0° to 180° (exclusive). Because trig functions repeat, other angles share the same value; the Inverse tab lists them within one full turn.
What is a reference angle?
It is the acute angle between the terminal side and the x-axis. Every trig value of θ equals the value for its reference angle up to a sign determined by the quadrant (All-Sine-Tangent-Cosine: all positive in I, sine in II, tangent in III, cosine in IV).
How accurate is the result?
The functions are evaluated in IEEE 754 double precision, about 15 to 16 significant digits. Values that are mathematically zero (such as cos 90°) are cleaned to exactly 0. Exact radical forms are shown for angles that are multiples of 30° or 45°.
What are hyperbolic functions?
sinh, cosh and tanh are built from exponentials: sinh x = (eˣ − e⁻ˣ)/2 and cosh x = (eˣ + e⁻ˣ)/2. They satisfy cosh²x − sinh²x = 1 and describe hanging cables (catenaries) and relativistic rapidity.
The unit-circle reason arccos(−1) is 180°: why cos inverse of −1 is 180°.
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