What are the inverse functions for?

Inverse sine, cosine and tangent, written as arcsin, arccos and arctan, find the angle that produces a given ratio. They answer the question, which angle has this sine?

Why is tangent undefined at 90 degrees?

Tangent is sine divided by cosine, and cosine is zero at 90 and 270 degrees, so dividing by zero leaves tangent undefined at those angles.

Sine Cosine Tangent Calculator

This calculator gives the three main trigonometric ratios, sine, cosine and tangent, for any angle, and also runs in reverse with the inverse functions to find an angle from a ratio. Sine, cosine and tangent describe the relationship between the angles and sides of a right triangle, summed up by the memory aid SOH CAH TOA: sine is the opposite side over the hypotenuse, cosine is the adjacent over the hypotenuse, and tangent is the opposite over the adjacent. These three ratios are the foundation of trigonometry and are used constantly in geometry, surveying, navigation, physics, engineering and computer graphics. You enter an angle, choose degrees or radians, and the calculator returns all three ratios at once, reporting tangent as undefined where cosine is zero, such as at 90 degrees. The second tool handles the reverse problem using arcsine, arccosine and arctangent: you enter a ratio and it returns the angle that produces it, in both degrees and radians, which is exactly what you need when you know two sides of a right triangle and want the angle between them. Sine and cosine accept ratios from minus one to one, while tangent accepts any value. Enter an angle or a ratio to see the results update as you type.

Angle to sin, cos, tan

Angle

Unit

sin0.707107

cos0.707107

tan1

Ratio to angle (inverse)

Function

Ratio value

45° (0.785398 rad)

How it works

For the angle, the calculator converts to radians if needed, then evaluates sine and cosine and divides them for tangent, cleaning up tiny rounding errors and reporting undefined where cosine is zero. For the inverse, it applies arcsine, arccosine or arctangent to your ratio and converts the resulting angle to both degrees and radians.

Worked example

At 45 degrees, sine and cosine are both about 0.7071 and tangent is exactly 1. Running the inverse, arctangent of 1 returns 45 degrees, which is pi over 4 radians.

Related calculators

Trig Calculator: all six functions.
Hypotenuse Calculator: right triangle sides.
Degrees to Radians Calculator: angle units.
Triangle Calculator: solve triangles.

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