Inverse Trig Calculator Guide
Learn when to use arcsin, arccos, and arctan to find angles from ratios.
Quick Answer
θ = asin(opposite / hypotenuse)When to Use This
Use an inverse trigonometric function when a side ratio is known and the angle is missing. Keep the ratio separate from a side length: entering 3 into asin is not equivalent to entering a 3 cm side together with a 5 cm hypotenuse.
Step-by-Step Method
- Form a ratio using two sides with matching units.
- Use asin for opposite/hypotenuse, acos for adjacent/hypotenuse or atan for opposite/adjacent.
- Choose degrees or radians for the returned angle.
- Substitute the angle into the original sine, cosine or tangent ratio.
Calculator Tips and Limits
- asin and acos accept real ratios only between -1 and 1, inclusive.
- The inverse function returns a principal angle, not every solution of a periodic trigonometric equation.
Worked Examples
A 3-to-5 side ratio
With opposite 3 cm and hypotenuse 5 cm, sin θ = 3/5 = 0.6. In degrees, asin(0.6) ≈ 36.86989765°, or 36.87° to two decimals. Checking sin(36.86989765°) gives approximately 0.6.
Tangent ratio one
If the opposite and adjacent legs are both 4 cm, tan θ = 4/4 = 1. atan(1) = 45° in DEG or π/4 in RAD.
Common Mistakes
Using a length as the ratio
asin(3) has no real result. Divide the two appropriate lengths first.
Assuming the principal value is the only angle
Within 0° to 180°, both 30° and 150° have sine 0.5. asin(0.5) returns 30°; solving a full equation requires the interval and symmetry.
FAQ
Is sin⁻¹ the same as 1/sin?
In inverse-function notation it means arcsine. The reciprocal of sine is cosecant, a different operation.
What if the opposite side is longer than the hypotenuse?
Those labels or measurements cannot describe the intended right triangle. Recheck them instead of forcing the ratio into range.
References and corrections
OpenStax: Algebra and Trigonometry 2e — further reading. The worked examples on this page are CalcSolver explanations. Read our calculation and correction method, or report an error with the input and expected result.