Key takeaways
- sin⁻¹(x), cos⁻¹(x), and tan⁻¹(x) turn a ratio back into an angle.
- sin⁻¹ and cos⁻¹ only accept x between -1 and 1; tan⁻¹ accepts any real number.
- sin⁻¹(x) is NOT the same as 1/sin(x) — the -1 means "inverse function", not "reciprocal".
- sin⁻¹(0.5) = 30° because sin(30°) = 0.5.
In this guide
What is an inverse trig function?
If sin(30°) = 0.5, then the inverse function reverses that: sin⁻¹(0.5) = 30°. Inverse trig functions (also called arcsine, arccosine, and arctangent) are how you find a missing angle once you already know a side ratio.
Why does inverse trig matter?
Regular sine, cosine, and tangent go from angle to ratio. But many real problems hand you the ratio first — a slope, a grade, a measured proportion — and ask you to find the angle. Inverse trig is the only way to reverse that direction.
A ramp with sine ratio 0.5 isn't obviously any particular angle just by looking at the number — inverse sine reveals it's exactly 30°.
The formula
| Symbol | What it means | Example |
|---|---|---|
| x | The ratio you already know | 0.5 |
| θ | The angle that produces that ratio, in degrees | 30° |
sin⁻¹(x) and cos⁻¹(x) only accept x between -1 and 1, since sine and cosine never go beyond that range. tan⁻¹(x) accepts any real number.
How to find the angle step by step
- 1Identify which ratio you have — is it from sine, cosine, or tangent?
- 2Check the value is valid. sin⁻¹ and cos⁻¹ only accept numbers from -1 to 1; tan⁻¹ accepts any number.
- 3Apply the matching inverse function to get the angle, typically in radians internally.
- 4Convert the result to degrees for an easier-to-read answer.
Enter any ratio between -1 and 1 below to see all three inverse angles worked out.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Confusing sin⁻¹(x) with 1/sin(x). The -1 here means "inverse function", not "reciprocal" — they're completely different operations.
- Forgetting there can be more than one angle with the same ratio. Inverse trig functions return one standard answer, but sine and cosine repeat their values at other angles too.
- Not converting the final radian result to degrees when a degree answer is expected.
Tips and tricks
- To go the other way — find a ratio from a known angle — use the sin, cos, tan calculator.
- Convert your final angle with the radians to degrees calculator if your tool returns radians.
- A ratio of exactly 0 always gives an angle of 0° for sin⁻¹ and tan⁻¹, and 90° for cos⁻¹.
Where you'll use it in real life
- Accessibility: finding a ramp's angle from its rise-to-length ratio.
- Roofing: converting a roof's pitch ratio into a workable angle.
- Navigation: recovering a bearing angle from measured forward and sideways distances.
- Physics: finding the angle of a force or incline from measured components.
Quick summary
Inverse trig functions reverse sine, cosine, and tangent: give them a ratio, and they hand back the angle. sin⁻¹ and cos⁻¹ only accept -1 to 1, while tan⁻¹ accepts anything. Try the calculator above with your own ratio.
Worked example
A wheelchair ramp rises 0.5 meters for every 1 meter of hypotenuse length, so its sine ratio is 0.5. What angle does the ramp make with the ground?
- 01x = 0.5
- 02sin⁻¹(0.5) = 30°
- 03cos⁻¹(0.5) = 60°
- 04tan⁻¹(0.5) = 26.5651°
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is sin⁻¹(x) used for?
It finds the angle whose sine equals x. For example, sin⁻¹(0.5) = 30°, because sin(30°) = 0.5.
Why do sin⁻¹ and cos⁻¹ need a value between -1 and 1?
Because sine and cosine only ever produce results between -1 and 1, so any ratio outside that range has no real angle behind it.
Does tan⁻¹ have the same -1 to 1 restriction?
No. Tangent can take any real value, so tan⁻¹(x) accepts any number as its input, not just -1 to 1.
Is sin⁻¹(x) the same as 1 ÷ sin(x)?
No — sin⁻¹(x) is the inverse function (it returns an angle), while 1 ÷ sin(x) is the reciprocal (it returns a ratio, also called cosecant).
