Inverse Trig Calculator – Find arcsin arccos arctan Online
Calculate inverse trigonometric functions including arcsin, arccos, and arctan with our free online inverse trig calculator. Get angle results in both degrees and radians.
Understanding Inverse Trigonometric Functions
Inverse trig functions answer the question: "What angle gives me this trig value?" If sin(30°) = 0.5, then arcsin(0.5) = 30°. They undo what the regular trig functions do.
There's a catch though. Sine, cosine, and tangent repeat their values, so there are infinitely many angles with the same trig value. To make the inverses proper functions, we restrict their ranges. Arcsin only returns angles between -90° and 90°. Arccos only returns angles between 0° and 180°. Arctan only returns angles between -90° and 90°.
You'll see inverse trig functions written two ways: arcsin(x) or sin⁻¹(x). Both mean the same thing. The ⁻¹ doesn't mean reciprocal – it means inverse function.
Domain and Range
arcsin(x)
Only accepts values from -1 to 1. Returns angles in quadrants I and IV.
arccos(x)
Only accepts values from -1 to 1. Returns angles in quadrants I and II.
arctan(x)
Accepts any real number. Returns angles in quadrants I and IV.
Common Inverse Trig Values
| x | arcsin(x) | arccos(x) | arctan(x) |
|---|---|---|---|
| -1 | -90° (-π/2) | 180° (π) | -45° (-π/4) |
| -√3/2 | -60° (-π/3) | 150° (5π/6) | — |
| -√2/2 | -45° (-π/4) | 135° (3π/4) | — |
| -1/2 | -30° (-π/6) | 120° (2π/3) | — |
| 0 | 0° (0) | 90° (π/2) | 0° (0) |
| 1/2 | 30° (π/6) | 60° (π/3) | — |
| √2/2 | 45° (π/4) | 45° (π/4) | — |
| √3/2 | 60° (π/3) | 30° (π/6) | — |
| 1 | 90° (π/2) | 0° (0) | 45° (π/4) |
| √3 | — | — | 60° (π/3) |
Worked Examples
sin(30°) = 0.5
arcsin(0.5) = 30° = π/6 rad
cos(180°) = -1
arccos(-1) = 180° = π rad
tan(45°) = 1
arctan(1) = 45° = π/4 rad
sin(-60°) = -√3/2
arcsin(-√3/2) = -60° = -π/3 rad
tan(60°) = √3
arctan(√3) = 60° = π/3 rad
Quick Fact
The notation sin⁻¹ for arcsin was introduced by John Herschel in 1813. He wanted a consistent notation for inverse functions. However, this causes confusion because sin²x means (sin x)², but sin⁻¹x doesn't mean 1/(sin x). Many mathematicians prefer "arcsin" to avoid this confusion.
Frequently Asked Questions
What's the difference between arcsin and 1/sin?
Completely different. arcsin(x) or sin⁻¹(x) is the inverse function – it gives you the angle. 1/sin(x) is the reciprocal, which is csc(x). The ⁻¹ in sin⁻¹ means "inverse function," not "reciprocal."
Why can't I calculate arcsin(2)?
Sine only outputs values between -1 and 1. No angle has a sine of 2. So arcsin(2) doesn't exist. Same for arccos of anything outside [-1, 1]. Arctan, though, accepts any real number.
Why does arccos give different answers than arcsin for the same input?
They have different restricted ranges. arcsin(0.5) = 30° but arccos(0.5) = 60°. Both sin(30°) and cos(60°) equal 0.5, but each inverse function returns angles from its own specific range.
When would I use inverse trig functions?
Anytime you know a trig ratio and need the angle. Finding the angle of a ramp given its slope. Calculating the launch angle of a projectile. Determining the angle in a triangle when you know side ratios.
What does sin⁻¹(x) mean?
sin⁻¹(x) means arcsin(x) – the inverse sine function. It's the angle whose sine is x. The notation is confusing because sin²(x) means (sin(x))², but sin⁻¹(x) doesn't mean 1/sin(x). It's just established notation.
Can inverse trig functions give negative angles?
Yes. arcsin and arctan both return negative angles for negative inputs. arcsin(-0.5) = -30°. arctan(-1) = -45°. Arccos never returns negative angles – its range is [0°, 180°].
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