Inverse Trigonometric Ratio Notes

Published on: September 29, 2026

This Inverse Trigonometric Ratio Calculator starts from two side lengths of a right triangle and works back to the angle, showing the working line by line. The ordinary ratios turn an angle into a number, so the inverse functions run that the other way and turn the number back into an angle. The ratio is formed first, and the inverse function is then applied to it. Following the same pattern every time makes it clear which two sides belong to which inverse function.

Step-by-step method

  1. Set up the problem.
  2. Write the inverse function for the chosen ratio.
  3. Substitute the two side lengths into the ratio.
  4. Divide to get the value of the ratio.
  5. Apply the inverse function to the ratio to get the angle.

Formulas:

\(\theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)\)
\(\theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)\)
\(\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)\)
See Example 1 Hide Example 1

Problem

\(\cos^{-1},\; \text{adjacent} = 8,\; \text{hypotenuse} = 17\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the inverse function for the chosen ratio.
  3. Substitute the two side lengths into the ratio.
  4. Divide to get the value of the ratio.
  5. Apply the inverse function to the ratio to get the angle.

Step 1

Step 1 - Set up the problem.

In this problem: The two side lengths that the arccosine needs are \(\text{adjacent} = 8\) and \(\text{hypotenuse} = 17\).

\(\text{adjacent} = 8,\; \text{hypotenuse} = 17\)

Step 2

Step 2 - Write the inverse function for the chosen ratio.

In this problem: The chosen function is arccosine, which turns the ratio of adjacent to hypotenuse back into an angle, giving \(\theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)\).

\(\theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)\)

Step 3

Step 3 - Substitute the two side lengths into the ratio.

In this problem: Replacing \(\text{adjacent}\) with \(8\) and \(\text{hypotenuse}\) with \(17\) gives \(\theta = \cos^{-1}\left(\frac{8}{17}\right)\).

\(\theta = \cos^{-1}\left(\frac{8}{17}\right)\)

Step 4

Step 4 - Divide to get the value of the ratio.

In this problem: Dividing \(8\) by \(17\) gives \(0.47\).

\(\theta = \cos^{-1}\left(0.47\right)\)

Step 5

Step 5 - Apply the inverse function to the ratio to get the angle.

In this problem: Taking the arccosine of \(0.47\) gives \(61.9275^{\circ}\).

\(\theta = \cos^{-1}\left(0.47\right) \approx 61.9275^{\circ}\)

Final Answer

\(\theta \approx 61.9275^{\circ}\)
See Example 2 Hide Example 2

Problem

\(\tan^{-1},\; \text{opposite} = 5,\; \text{adjacent} = 12\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the inverse function for the chosen ratio.
  3. Substitute the two side lengths into the ratio.
  4. Divide to get the value of the ratio.
  5. Apply the inverse function to the ratio to get the angle.

Step 1

Step 1 - Set up the problem.

In this problem: The two side lengths that the arctangent needs are \(\text{opposite} = 5\) and \(\text{adjacent} = 12\).

\(\text{opposite} = 5,\; \text{adjacent} = 12\)

Step 2

Step 2 - Write the inverse function for the chosen ratio.

In this problem: The chosen function is arctangent, which turns the ratio of opposite to adjacent back into an angle, giving \(\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)\).

\(\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)\)

Step 3

Step 3 - Substitute the two side lengths into the ratio.

In this problem: Replacing \(\text{opposite}\) with \(5\) and \(\text{adjacent}\) with \(12\) gives \(\theta = \tan^{-1}\left(\frac{5}{12}\right)\).

\(\theta = \tan^{-1}\left(\frac{5}{12}\right)\)

Step 4

Step 4 - Divide to get the value of the ratio.

In this problem: Dividing \(5\) by \(12\) gives \(0.42\).

\(\theta = \tan^{-1}\left(0.42\right)\)

Step 5

Step 5 - Apply the inverse function to the ratio to get the angle.

In this problem: Taking the arctangent of \(0.42\) gives \(22.6199^{\circ}\).

\(\theta = \tan^{-1}\left(0.42\right) \approx 22.6199^{\circ}\)

Final Answer

\(\theta \approx 22.6199^{\circ}\)
See Example 3 Hide Example 3

Problem

\(\sin^{-1},\; \text{opposite} = 3,\; \text{hypotenuse} = 5\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the inverse function for the chosen ratio.
  3. Substitute the two side lengths into the ratio.
  4. Divide to get the value of the ratio.
  5. Apply the inverse function to the ratio to get the angle.

Step 1

Step 1 - Set up the problem.

In this problem: The two side lengths that the arcsine needs are \(\text{opposite} = 3\) and \(\text{hypotenuse} = 5\).

\(\text{opposite} = 3,\; \text{hypotenuse} = 5\)

Step 2

Step 2 - Write the inverse function for the chosen ratio.

In this problem: The chosen function is arcsine, which turns the ratio of opposite to hypotenuse back into an angle, giving \(\theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)\).

\(\theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)\)

Step 3

Step 3 - Substitute the two side lengths into the ratio.

In this problem: Replacing \(\text{opposite}\) with \(3\) and \(\text{hypotenuse}\) with \(5\) gives \(\theta = \sin^{-1}\left(\frac{3}{5}\right)\).

\(\theta = \sin^{-1}\left(\frac{3}{5}\right)\)

Step 4

Step 4 - Divide to get the value of the ratio.

In this problem: Dividing \(3\) by \(5\) gives \(0.6\).

\(\theta = \sin^{-1}\left(0.6\right)\)

Step 5

Step 5 - Apply the inverse function to the ratio to get the angle.

In this problem: Taking the arcsine of \(0.6\) gives \(36.8699^{\circ}\).

\(\theta = \sin^{-1}\left(0.6\right) \approx 36.8699^{\circ}\)

Final Answer

\(\theta \approx 36.8699^{\circ}\)
See Example 4 Hide Example 4

Problem

\(\cos^{-1},\; \text{adjacent} = 5,\; \text{hypotenuse} = 13\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the inverse function for the chosen ratio.
  3. Substitute the two side lengths into the ratio.
  4. Divide to get the value of the ratio.
  5. Apply the inverse function to the ratio to get the angle.

Step 1

Step 1 - Set up the problem.

In this problem: The two side lengths that the arccosine needs are \(\text{adjacent} = 5\) and \(\text{hypotenuse} = 13\).

\(\text{adjacent} = 5,\; \text{hypotenuse} = 13\)

Step 2

Step 2 - Write the inverse function for the chosen ratio.

In this problem: The chosen function is arccosine, which turns the ratio of adjacent to hypotenuse back into an angle, giving \(\theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)\).

\(\theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)\)

Step 3

Step 3 - Substitute the two side lengths into the ratio.

In this problem: Replacing \(\text{adjacent}\) with \(5\) and \(\text{hypotenuse}\) with \(13\) gives \(\theta = \cos^{-1}\left(\frac{5}{13}\right)\).

\(\theta = \cos^{-1}\left(\frac{5}{13}\right)\)

Step 4

Step 4 - Divide to get the value of the ratio.

In this problem: Dividing \(5\) by \(13\) gives \(0.38\).

\(\theta = \cos^{-1}\left(0.38\right)\)

Step 5

Step 5 - Apply the inverse function to the ratio to get the angle.

In this problem: Taking the arccosine of \(0.38\) gives \(67.3801^{\circ}\).

\(\theta = \cos^{-1}\left(0.38\right) \approx 67.3801^{\circ}\)

Final Answer

\(\theta \approx 67.3801^{\circ}\)