Trig Function Calculator
Enter any angle and choose your unit (degrees, radians, or gradians). The calculator instantly returns all six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. Switch to Inverse mode to recover the angle from a known function value. The worked steps panel shows every calculation so you can follow along.
The six trigonometric functions
All six trig functions relate the angles of a right triangle to the ratios of its sides. For an acute angle in a right triangle: - Sine (sin) = opposite / hypotenuse - Cosine (cos) = adjacent / hypotenuse - Tangent (tan) = opposite / adjacent - Cosecant (csc) = hypotenuse / opposite = 1 / sin - Secant (sec) = hypotenuse / adjacent = 1 / cos - Cotangent (cot) = adjacent / opposite = 1 / tan The mnemonic SOHCAHTOA helps you remember sin, cos, and tan. The remaining three (csc, sec, cot) are simply the reciprocals. These definitions extend beyond right triangles to all real angles via the unit circle, which is why you can compute sin(270 degrees) or sin(-45 degrees).
Degrees, radians, and gradians
There are three common angle units. Degrees split a full rotation into 360 equal parts and are most familiar from everyday use. Radians are the SI standard: a full circle equals 2 pi radians (approximately 6.2832 rad), and a right angle is pi/2 rad. Gradians (also called grads or gon) split a full circle into 400 parts, so a right angle is exactly 100 grad. The conversion formulas are: radians = degrees x (pi / 180), gradians = degrees x (400 / 360). This calculator accepts all three units and converts internally before computing.
Inverse trigonometric functions
The inverse functions - arcsin, arccos, and arctan - reverse the process. Instead of entering an angle and getting a ratio, you enter a ratio and get the angle. For example, arcsin(0.5) = 30 degrees (or pi/6 radians) because sin(30 degrees) = 0.5. Because trig functions are periodic, there are infinitely many angles with the same sine. Calculators return the principal value: arcsin returns angles in the range -90 to 90 degrees, arccos returns 0 to 180 degrees, and arctan returns -90 to 90 degrees. Use Switch mode at the top of this calculator to access inverse trig.
Key identities and relationships
Several identities connect the six functions. The Pythagorean identity sin^2(x) + cos^2(x) = 1 is the most fundamental, and it follows directly from the Pythagorean theorem applied to the unit circle. Two related forms are 1 + tan^2(x) = sec^2(x) and 1 + cot^2(x) = csc^2(x). Parity identities state that sin and tan are odd functions (sin(-x) = -sin(x)), while cos is even (cos(-x) = cos(x)). Complementary-angle identities show that sin(90 deg - x) = cos(x), so sine and cosine are co-functions - as are tan/cot and sec/csc. This is where the co- prefix in cosine, cosecant, and cotangent originates.
Common angle trig values
| Angle (degrees) | Angle (radians) | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 0 | undef. | 1 | undef. |
| 30 | π/6 | 1/2 | √3/2 | 1/√3 | 2 | 2/√3 | √3 |
| 45 | π/4 | √2/2 | √2/2 | 1 | √2 | √2 | 1 |
| 60 | π/3 | √3/2 | 1/2 | √3 | 2/√3 | 2 | 1/√3 |
| 90 | π/2 | 1 | 0 | undef. | 1 | undef. | 0 |
| 120 | 2π/3 | √3/2 | -1/2 | -√3 | 2/√3 | -2 | -1/√3 |
| 135 | 3π/4 | √2/2 | -√2/2 | -1 | √2 | -√2 | -1 |
| 150 | 5π/6 | 1/2 | -√3/2 | -1/√3 | 2 | -2/√3 | -√3 |
| 180 | π | 0 | -1 | 0 | undef. | -1 | undef. |
| 270 | 3π/2 | -1 | 0 | undef. | -1 | undef. | 0 |
| 360 | 2π | 0 | 1 | 0 | undef. | 1 | undef. |
Exact values of the six trig functions at frequently tested angles.
Frequently asked questions
What is the difference between sin and arcsin?
Sin takes an angle as input and returns a ratio (a number between -1 and 1). Arcsin does the reverse: it takes a ratio and returns the angle. For example, sin(30 degrees) = 0.5, while arcsin(0.5) = 30 degrees. Use forward mode for sin, and switch to inverse mode for arcsin.
Why is tan undefined at 90 degrees and 270 degrees?
Tan = sin / cos. At 90 degrees and 270 degrees, cos equals exactly zero, so the division is undefined. The same applies to sec (= 1/cos) at those angles. Similarly, csc and cot are undefined wherever sin = 0, which is at 0, 180, and 360 degrees.
How do I convert degrees to radians?
Multiply degrees by pi/180. For example, 45 degrees x (pi/180) = pi/4 rad, which is approximately 0.7854 rad. To go the other way, multiply radians by 180/pi. This calculator handles the conversion automatically when you choose the angle unit.
What does the Pythagorean identity mean?
The identity sin^2(x) + cos^2(x) = 1 says that for any angle, the squares of sin and cos always add up to exactly 1. It comes from the Pythagorean theorem applied to a point on the unit circle (a circle of radius 1 centred at the origin). Because the radius is 1, opposite^2 + adjacent^2 = 1^2 = 1, which gives sin^2 + cos^2 = 1.
What is the range of the inverse trig functions?
Arcsin returns angles from -90 to 90 degrees (-pi/2 to pi/2 rad). Arccos returns 0 to 180 degrees (0 to pi rad). Arctan returns -90 to 90 degrees (-pi/2 to pi/2 rad). These are the principal values. Because trig functions repeat every period, there are always other angles that give the same result, but by convention the inverse functions return only the angle in the principal range.