Trigonometric Functions Calculator
Enter any angle to instantly compute all six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. Choose degrees, radians, or gradians. Switch to inverse mode to find the angle from a ratio. The steps panel shows the full working, and the reference table lists the exact values for the most common special angles.
What are the six trigonometric functions?
Trigonometry defines six ratios between the sides of a right triangle, each named after its relationship to the angle. For an acute angle in a right triangle, the opposite side faces the angle, the adjacent side is the remaining non-hypotenuse leg, and the hypotenuse is the longest side. Sine (sin) is opposite over hypotenuse; cosine (cos) is adjacent over hypotenuse; tangent (tan) is opposite over adjacent. The other three are reciprocals: cosecant (csc) is 1/sin, secant (sec) is 1/cos, and cotangent (cot) is 1/tan (or cos/sin). These definitions extend beyond right triangles to all real angles through the unit circle, which is why they are also called circular functions.
Degrees, radians, and gradians
Angles can be measured in three common units. A full circle is 360 degrees, 2pi radians, or 400 gradians. Radians are the SI unit and are used in almost all mathematical and scientific formulas because they make calculus identities cleaner: d/dx sin(x) = cos(x) holds only when x is in radians. Gradians split the circle into 400 parts and were historically used in surveying. To convert, use: 1 degree = pi/180 radians = 10/9 gradians. This calculator accepts any of the three units and converts internally before computing.
The unit circle and periodic behavior
The unit circle is a circle of radius 1 centered at the origin of a coordinate plane. Any point on it can be described as (cos(theta), sin(theta)) for an angle theta measured counter-clockwise from the positive x-axis. This shows directly that sin and cos always lie between -1 and 1, and that they repeat every 360 degrees (2pi radians). Tangent repeats every 180 degrees (pi radians) and is undefined wherever cos = 0, at 90, 270, 450... degrees. Cosecant and secant are undefined wherever sin = 0 or cos = 0, respectively. The Pythagorean identity sin^2(theta) + cos^2(theta) = 1 follows immediately from the unit circle definition.
Inverse trigonometric functions and their domains
Inverse functions undo the forward ones: arcsin(x) returns the angle whose sine is x. Because sin repeats, its inverse is only well-defined on a restricted interval. The principal value ranges are: arcsin returns -90 to 90 degrees; arccos returns 0 to 180 degrees; arctan returns -90 to 90 degrees; arccsc and arcsec follow from their definitions via 1/ratio; arccot returns 0 to 180 degrees. Domain restrictions matter: arcsin and arccos only accept inputs from -1 to 1, while arccsc and arcsec require an absolute value of at least 1. This calculator reports "undefined" for out-of-domain inputs rather than giving a wrong answer.
Special angles: exact values
| Angle (°) | Angle (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 |
| 120 | 2π/3 | √3/2 | -1/2 | -√3 | 2√3/3 | -2 | -√3/3 |
| 135 | 3π/4 | √2/2 | -√2/2 | -1 | √2 | -√2 | -1 |
| 150 | 5π/6 | 1/2 | -√3/2 | -√3/3 | 2 | -2√3/3 | -√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. |
Commonly memorized exact values. "undef." means the function is undefined at that angle (division by zero).
Frequently asked questions
What is SOHCAHTOA?
SOHCAHTOA is a mnemonic for the three primary trigonometric ratios in a right triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. It covers sin, cos, and tan. The reciprocal functions csc, sec, and cot are 1/sin, 1/cos, and 1/tan respectively.
When is tangent undefined?
Tangent is undefined whenever cos(angle) = 0, because tan = sin/cos and division by zero is undefined. This happens at 90 degrees, 270 degrees, and any multiple of 180 degrees offset by 90 (i.e., 90 + 180n for any integer n). At those angles the tangent line to the unit circle is vertical, which has no finite slope.
Why do sin and cos always stay between -1 and 1?
On the unit circle, (cos(theta), sin(theta)) is always a point on a circle of radius 1. No coordinate of a point on that circle can exceed 1 or be less than -1. So sin and cos are always in the range [-1, 1]. Cosecant and secant, being reciprocals, therefore have absolute values of 1 or greater.
How do I convert between degrees and radians?
Multiply degrees by pi/180 to get radians. Multiply radians by 180/pi to get degrees. Key landmarks: 0 degrees = 0 rad, 30 degrees = pi/6 rad, 45 degrees = pi/4 rad, 60 degrees = pi/3 rad, 90 degrees = pi/2 rad, 180 degrees = pi rad, 360 degrees = 2*pi rad. The calculator handles the conversion automatically.
What is the Pythagorean identity?
The Pythagorean identity states that sin^2(theta) + cos^2(theta) = 1 for every angle theta. It comes directly from the unit circle definition. Dividing both sides by cos^2 gives 1 + tan^2 = sec^2, and dividing by sin^2 gives cot^2 + 1 = csc^2. These three forms are all called Pythagorean identities.