How to use the trigonometry calculator
- Choose Angle to sin, cos, tan, Value to angle (inverse) or Solve a right triangle.
- For the first mode, enter an angle in degrees or radians. Negative angles and angles above 360 are fine.
- For the inverse mode, pick arcsin, arccos or arctan and enter a value. For a right triangle, enter one acute angle, say which side you know and its length.
- Read the results, including exact values for special angles such as 30, 45 and 60 degrees.
Formula
sin = opposite / hypotenuse, cos = adjacent / hypotenuse, tan = opposite / adjacent
csc = 1 / sin, sec = 1 / cos, cot = 1 / tan
radians = degrees x pi / 180, degrees = radians x 180 / pi
Inverse functions return principal values: arcsin and arctan from -90 to 90 degrees, arccos from 0 to 180 degrees
The angle is first reduced to the range 0 to 360 degrees, so exact multiples of 90 degrees give exact values. Results closer to zero than 1e-12 are shown as 0 to hide floating-point noise.
Worked example
At 30 degrees, sin is 0.5 (exactly 1/2), cos is 0.8660254038 (exactly sqrt(3)/2) and tan is 0.5773502692. At 60 degrees cos is exactly 1/2, and at 45 degrees tan is exactly 1.
At 90 degrees, tan and sec are undefined because cosine is zero. At 180 degrees sin is exactly 0, and at -30 degrees sin is -0.5 in quadrant IV. An angle of 750 degrees has the coterminal angle 30 degrees.
Inverse: arcsin(0.5) is 30 degrees, arccos(0) is 90 degrees and arctan(1) is 45 degrees. Right triangle: with a 35 degree angle and a hypotenuse of 10, the opposite side is 5.73576436 and the adjacent side is 8.19152044.
Degrees and radians
A full turn is 360 degrees or 2 pi radians. Most school problems use degrees, while calculus and programming use radians. If you enter radians, the tool shows the equivalent degrees and the coterminal angle between 0 and 360, so you can check which quadrant you are in. A scientific calculator such as the scientific calculator can chain these functions into longer expressions.
Undefined values and domains
Some inputs have no answer, and the calculator says so plainly.
- Tangent and secant are undefined where cosine is zero, at 90 and 270 degrees.
- Cosecant and cotangent are undefined where sine is zero, at 0 and 180 degrees.
- Arcsin and arccos accept only values from -1 to 1, because sine and cosine never go outside that range.
- In a right triangle the entered acute angle must be strictly between 0 and 90 degrees.
Related tools
To solve any triangle, not just a right one, use the triangle calculator. To find the missing side of a right triangle from two sides, the Pythagorean theorem calculator is the direct route.
Assumptions and limits
- Trig functions follow the standard unit-circle definitions with the angle measured counterclockwise from the positive x-axis.
- Exact values are shown only for angles whose reference angle is 0, 30, 45, 60 or 90 degrees.
- Inverse functions return the principal value. Other angles with the same ratio exist and are described in the steps.
- Angles up to 1,000,000,000 degrees are accepted. Right triangle lengths up to 10^12.
- Everything runs in your browser. Nothing you enter is sent anywhere.
Frequently asked questions
What are sin, cos and tan?
They are ratios of sides in a right triangle for a given acute angle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse and tangent is opposite over adjacent. They extend to all angles through the unit circle.
Why is tan 90 degrees undefined?
Tangent equals sine divided by cosine, and cosine of 90 degrees is exactly zero. Division by zero has no value, so the calculator reports undefined instead of a huge number.
How do I convert degrees to radians?
Multiply by pi and divide by 180. For example, 30 degrees is pi/6, about 0.5235987756 radians. To go back, multiply by 180 and divide by pi.
What does arcsin mean?
It is the inverse of sine: it gives the angle whose sine is a given value. Because many angles share one sine, the calculator returns the principal angle between -90 and 90 degrees.
How do I solve a right triangle with one angle and one side?
Use sine, cosine or tangent to find the other sides. If you know the hypotenuse h and angle t, opposite is h sin t and adjacent is h cos t. The calculator does this for any of the three known sides.