Inverse Trig Calculator
Calculate inverse trigonometric functions (arcsin, arccos, arctan)
How It Works
Overview
An inverse trig calculator answers the question, "I know the ratio — what's the angle?" While sin, cos, and tan turn an angle into a ratio, their inverses (arcsin, arccos, arctan and the reciprocal versions) turn a ratio back into an angle. The calculator returns the answer in both degrees and radians, plus the underlying expression.
Inverse trig functions are the workhorses of any problem where you're given two sides of a right triangle and need the angle: bevel cuts, slope angles, vector directions, angle of arrival in physics, or converting Cartesian coordinates to polar. Because each trig function is periodic, the inverses are restricted to a single "principal branch" so they output a unique answer.
The Formula
Each inverse has a domain (allowed inputs) and a principal range:
- arcsin(x): domain [−1, 1], range [−90°, 90°]
- arccos(x): domain [−1, 1], range [0°, 180°]
- arctan(x): domain all real x, range (−90°, 90°)
- arccsc(x), arcsec(x): domain |x| ≥ 1
- arccot(x): domain all real x
These restrictions guarantee a one-to-one mapping. If your problem needs an angle outside the principal range, use symmetry — for example, sin is also positive in the second quadrant, so the second solution to sin θ = 0.5 is 180° − 30° = 150°.
Worked Example
A ramp rises 2 ft over a horizontal run of 12 ft. What's the ramp angle?
- tan θ = rise / run = 2 / 12 = 0.16667
- θ = arctan(0.16667) ≈ 9.46° (≈ 0.1651 rad)
Sanity check: an ADA-compliant ramp must be no steeper than 1:12, which is 4.76°. A 1:6 ramp at 9.46° is too steep for accessibility but fine for many other uses. Now suppose you measure 0.5 for the sine of an angle: arcsin(0.5) = 30° exactly, and arccos(0.5) = 60°.
When to Use This
- Solving right triangles — given two sides, recover the missing angles.
- Converting (x, y) to polar (r, θ) — use arctan(y/x), or atan2 in code, to find θ.
- Slopes and grades — the angle of a hill given its rise and run, or a roof angle from pitch.
- Physics — finding launch angles for projectiles, refraction angles via Snell's law, or angles of incidence in optics.
- CAD and CNC — programming a cutter to follow a path defined by coordinates rather than angles.
Common Mistakes to Avoid
- Inputting an out-of-domain value. arcsin(1.5) is undefined. If you computed a "sine" greater than 1 from a triangle, recheck your sides — you probably divided by the wrong leg.
- Confusing sin⁻¹ with 1/sin. The −1 superscript is functional inverse notation, not an exponent. 1/sin(θ) is csc(θ).
- Ignoring the quadrant. arctan(y/x) loses sign info when x is negative. Use atan2(y, x) in code to get the right quadrant from -180° to 180°.
- Reading the calculator in the wrong unit. 0.5236 rad and 30° are the same angle — confirm whether your downstream formula expects degrees or radians.
- Assuming a unique answer. If you need every angle satisfying sin θ = 0.5, add the supplementary 180° − 30° = 150° plus full 360° rotations.
Frequently Asked Questions
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