Calculators / Trigonometry / Inverse Trigonometric Ratio Calculator

Inverse Trigonometric Ratio Calculator

Published on: January 5, 2025

This Inverse Trigonometric Ratio Calculator helps you find an angle in a right triangle using arcsine, arccosine, and arctangent. It uses the formulas sin⁻¹(opposite / hypotenuse), cos⁻¹(adjacent / hypotenuse), and tan⁻¹(opposite / adjacent) to calculate the angle from the given side values. Choose the inverse function you need, enter the correct two sides, and the calculator will work out the angle. It is a simple way to check answers, understand inverse trigonometric ratios, and practise basic trigonometry step by step.

Step-by-step method

  1. Identify what is given.
  2. Form the ratio.
  3. Apply the inverse function to find the angle.

Formulas:

Formula: sin⁻¹

θ=sin⁻¹(
opposite
hypotenuse
)

Formula: cos⁻¹

θ=cos⁻¹(
adjacent
hypotenuse
)

Formula: tan⁻¹

θ=tan⁻¹(
opposite
adjacent
)

Example 1: sin⁻¹ with opposite = 3, hypotenuse = 5

Step 1 - Identify what is given.

In this problem: The given values are opposite = 3 and hypotenuse = 5.

opposite=3hypotenuse=5

Step 2 - Form the ratio.

In this problem: Form the ratio: opposite ÷ hypotenuse = 3 ÷ 5 = 0.6.

3
5
=0.6

Step 3 - Apply the inverse function to find the angle.

In this problem: Apply the inverse function: θ = sin⁻¹(0.6) = 36.86989765°.

θ=sin⁻¹(0.6)=36.86989765°

Final answer: θ = 36.86989765°

Example 2: tan⁻¹ with opposite = 3, adjacent = 4

Step 1 - Identify what is given.

In this problem: The given values are opposite = 3 and adjacent = 4.

opposite=3adjacent=4

Step 2 - Form the ratio.

In this problem: Form the ratio: opposite ÷ adjacent = 3 ÷ 4 = 0.75.

3
4
=0.75

Step 3 - Apply the inverse function to find the angle.

In this problem: Apply the inverse function: θ = tan⁻¹(0.75) = 36.86989765°.

θ=tan⁻¹(0.75)=36.86989765°

Final answer: θ = 36.86989765°