Exact Value of Trig Functions Calculator
Enter any angle and get sin, cos, tan, csc, sec, cot as exact radical/fraction forms — e.g. sin(30°) = 1/2, cos(45°) = √2/2, tan(60°) = √3 — plus decimal approximations and a unit-circle diagram.
Find exact trig values
Tip: in radian mode you can type π-expressions like pi/6, 2*pi/3 or 3pi/2.
Special angles — tap one
Complete exact-value reference table — 48 special angles
Every special angle with a neat radical form, all six functions. Generated from the same verified engine as the calculator above.
FAQ — exact trig values
What is an exact trig value?
An exact value writes the result with radicals and fractions instead of a rounded decimal. For example, sin(30°) = 1/2, cos(45°) = √2/2, and tan(60°) = √3. Exact forms never lose precision to rounding.
Which angles have exact values?
The special angles: 0°, 30°, 45°, 60°, 90° and their equivalents in all four quadrants (120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°), plus the 15°/75°, 18°/36°/54°/72°, and 22.5°/67.5° families — 48 angles in total, all covered in the reference table above. Even tricky ones work: tan(18°) = √(25−10√5)/5.
What is the exact value of sin(45°)?
sin(45°) = √2/2, which is approximately 0.7071067812. It comes from the isosceles right triangle with legs 1, 1 and hypotenuse √2.
Why is tan(90°) undefined?
Because tan(90°) = sin(90°)/cos(90°) = 1/0, and division by zero is undefined. The same happens for cot(0°), csc(0°), and sec(90°) — anywhere the denominator function equals zero.
How do I find exact values without a calculator?
Use three steps: 1) reduce the angle into 0°–360°; 2) find its reference angle (for example, 150° has reference angle 30°); 3) apply the ASTC sign rule for the quadrant. Example: cos(150°) = −cos(30°) = −√3/2, because cosine is negative in quadrant II.
When should I use exact values instead of decimals?
Use exact forms in algebra, proofs, and exams where rounding would compound errors. Use decimals when you need a numeric approximation for measurement or engineering. This page gives you both, side by side.
Free tool by NRTechStudio