Exact Value of Trig Functions Calculator
Select a standard angle (0° to 360° in standard unit-circle increments) and instantly see all six trigonometric functions in their exact radical form and as six-decimal-place approximations.
Angle
Exact form shown in the table below
How does this calculator work?
Standard angles (multiples of 30° and 45°) have trig values expressible in exact radical form from two special triangles. sin, cos, tan, csc, sec and cot are shown both exactly (e.g. √3/2) and as decimals. Functions are undefined where their denominator (sin or cos) equals zero.
Formula
How this is calculated
The standard angles — multiples of 30° and 45° — have trigonometric values that can be expressed exactly as fractions involving √2 and √3. These come from the geometry of two special right triangles: the 30-60-90 triangle (sides 1 : √3 : 2) and the 45-45-90 triangle (sides 1 : 1 : √2). Applying SOHCAHTOA to each triangle, then extending to the full circle using the sign rules for each quadrant, produces the complete table of exact values.
The reciprocal functions follow directly: cscθ = 1/sinθ, secθ = 1/cosθ, and cotθ = cosθ/sinθ. Wherever sinθ = 0 (0°, 180°, 360°) csc and cot are undefined, and wherever cosθ = 0 (90°, 270°) sec and tan are undefined; those entries show —.
Exact forms matter in algebra and calculus because they avoid rounding errors that accumulate in subsequent calculations. For example, simplifying sin(45°)² + cos(45°)² with the exact value √2/2 immediately gives 1/2 + 1/2 = 1, whereas a rounded decimal would give 0.999… after squaring and adding.
Frequently asked questions
The 45-45-90 triangle (isosceles right triangle) has leg:hypotenuse ratio 1:√2, giving √2/2 for sin and cos of 45°. The 30-60-90 triangle (half of an equilateral triangle) has sides 1:√3:2, giving 1/2 for sin 30° and √3/2 for sin 60°. All other standard-angle values derive from these two triangles.
Use reference angles and quadrant signs. 150° lies in quadrant II with reference angle 30°, so sin 150° = +sin 30° = 1/2 (sin is positive in Q2) and cos 150° = -cos 30° = -√3/2 (cos is negative in Q2). Apply the ASTC rule (All, Sin, Tan, Cos positive in Q1, Q2, Q3, Q4 respectively).
tanθ and secθ are undefined when cosθ = 0, i.e. at 90° and 270°. cscθ and cotθ are undefined when sinθ = 0, i.e. at 0°, 180°, and 360°. Division by zero on the unit circle corresponds to the angle being on an axis where the denominator function crosses zero.
TG we-Calculate Editorial Team. (2026). Exact Value of Trig Functions Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/exact-value-of-trig-functions-calculator
TG we-Calculate Editorial Team. "Exact Value of Trig Functions Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/exact-value-of-trig-functions-calculator.
TG we-Calculate Editorial Team, "Exact Value of Trig Functions Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/exact-value-of-trig-functions-calculator
@misc{wecalculate_exact_value_of_trig_functions_calculator, title = {Exact Value of Trig Functions Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/exact-value-of-trig-functions-calculator}}, year = {2026}, note = {TG we-Calculate} }
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