Reverse the function, not the fraction
If sin(θ) = r, the inverse sine function asks for an angle with that sine ratio. The notation sin⁻¹ often means arcsine; it does not mean 1/sin(θ), which is cosecant.
Use asin for an opposite/hypotenuse ratio, acos for adjacent/hypotenuse, and atan for opposite/adjacent when working with acute right-triangle angles.
Understand principal values
Trigonometric functions repeat, so a ratio can correspond to many angles. Calculators choose one principal value: asin returns from −90° to 90°, acos from 0° to 180°, and atan strictly between −90° and 90°.
A general trigonometric equation may require other angles and full-turn adjustments. One calculator result is not automatically a complete solution set.
Check input domains and output units
For real results, asin and acos need a ratio between −1 and 1. Atan accepts any finite real ratio. A length ratio greater than one cannot represent opposite/hypotenuse in a right triangle.
Choose the preferred output unit on the inverse-trig page. It displays both degrees and radians so you can check that a conversion was not mistaken for a different answer.
LET’S WORK IT THROUGH
An acute angle has opposite side 6 and hypotenuse 12. Find the angle.
- The sine ratio is 6/12 = 0.5.
- Compute asin(0.5).
- In degrees, the principal angle is 30°.
THE ANSWERThe angle is 30°, or π/6 radians.
YOUR TURN
A small check for understanding.
What is atan(1) in degrees?
Reveal the answer
45°, the principal angle whose tangent is 1.
Now try your own numbers.
The inverse trigonometry includes input guidance and the limits of its supported calculations. Use it alongside the method above.
Open Inverse trigonometry ↗A note on precision: preserve exact fractions where possible and round numerical results at the end. Found an unclear step? Prepare a correction report.