Trigonometry · Quick Reference

Inverse Trigonometric Functions Cheat Sheet

Trigonometry · Lesson 5/8

In one line: inverse trig functions reverse the process of sine, cosine, and tangent — instead of giving a ratio from an angle, they give back the angle from a ratio.

Key Ideas

1Inverse Sine (arcsin or sin^-1). If sin(theta) = x, then arcsin(x) = theta. It answers: 'what angle has this sine value?'

2Inverse Cosine (arccos or cos^-1). If cos(theta) = x, then arccos(x) = theta.

3Inverse Tangent (arctan or tan^-1). If tan(theta) = x, then arctan(x) = theta.

4Restricted Ranges. Since sin, cos, and tan repeat, their inverses are restricted to specific output ranges to give exactly one answer per input (e.g., arcsin outputs between -90° and 90°).

5Using Inverse Trig to Solve Triangles. Given a ratio of two sides, use the matching inverse trig function to find the unknown angle.

Worked Examples

Find arcsin(0.5).

30°

Find arccos(0.5).

60°

A right triangle has opposite side 5 and hypotenuse 10. Find angle A using inverse sine.

A = 30°

Formulas

If sin(theta)=x, theta = arcsin(x)

If cos(theta)=x, theta = arccos(x)

If tan(theta)=x, theta = arctan(x)

Practice Yourself

Find arcsin(1).

90°

Find arccos(1).

0°

Find arctan(1).

45°