01 Key Concepts
Inverse Sine (arcsin or sin^-1)
If sin(theta) = x, then arcsin(x) = theta. It answers: 'what angle has this sine value?'
Inverse Cosine (arccos or cos^-1)
If cos(theta) = x, then arccos(x) = theta.
Inverse Tangent (arctan or tan^-1)
If tan(theta) = x, then arctan(x) = theta.
Restricted 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°).
Using Inverse Trig to Solve Triangles
Given a ratio of two sides, use the matching inverse trig function to find the unknown angle.
02 Key Formulas
- If sin(theta)=x, theta = arcsin(x)
- If cos(theta)=x, theta = arccos(x)
- If tan(theta)=x, theta = arctan(x)
03 Solved Examples
Example 1 Find arcsin(0.5).
- Ask: what angle has a sine of 0.5?
- sin(30°) = 0.5.
Answer: 30°
Example 2 Find arccos(0.5).
- Ask: what angle has a cosine of 0.5?
- cos(60°) = 0.5.
Answer: 60°
Example 3 A right triangle has opposite side 5 and hypotenuse 10. Find angle A using inverse sine.
- sin(A) = opposite/hypotenuse = 5/10 = 0.5.
- A = arcsin(0.5).
Answer: A = 30°
04 Practice Questions
1Find arcsin(1).
90°
2Find arccos(1).
0°
3Find arctan(1).
45°
4A triangle has opposite 3, hypotenuse 6. Find angle A using arcsin.
arcsin(0.5)=30°
5What does arccos(x) answer?
The angle whose cosine is x
📄 Inverse Trigonometric Functions — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.