Inverse Trigonometric Ratio Calculator

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:

Example 1:

Step 1 - Identify what is given.

In this problem: The given values are \(\text{opposite} = 3\) and \(\text{hypotenuse} = 5\).

Step 2 - Form the ratio.

In this problem: Form the ratio: \(\frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} = 0.6\).

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

In this problem: Apply the inverse function: \(\theta = \sin^{-1}(0.6) \approx 36.8699^{\circ}\).

Final answer:

Example 2:

Step 1 - Identify what is given.

In this problem: The given values are \(\text{opposite} = 3\) and \(\text{adjacent} = 4\).

Step 2 - Form the ratio.

In this problem: Form the ratio: \(\frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4} = 0.75\).

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

In this problem: Apply the inverse function: \(\theta = \tan^{-1}(0.75) \approx 36.8699^{\circ}\).

Final answer:

See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

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

Step 1

Step 1 - Identify what is given.

In this problem: The given values are \(\text{opposite} = 3\) and \(\text{hypotenuse} = 5\).

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

Step 2

Step 2 - Form the ratio.

In this problem: Form the ratio: \(\frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} = 0.6\).

\(\frac{3}{5} = 0.6\)

Step 3

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

In this problem: Apply the inverse function: \(\theta = \sin^{-1}(0.6) \approx 36.8699^{\circ}\).

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

Final Answer

\(\sin^{-1} \text{ with } \text{opposite} = 3,\; \text{hypotenuse} = 5 = \theta \approx 36.8699^{\circ}\)

See Example 2 Hide Example 2

Problem

\(\tan^{-1} \text{ with } \text{opposite} = 3,\; \text{adjacent} = 4\)

Approach

Step-by-step method

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

Step 1

Step 1 - Identify what is given.

In this problem: The given values are \(\text{opposite} = 3\) and \(\text{adjacent} = 4\).

\(\text{opposite} = 3,\; \text{adjacent} = 4\)

Step 2

Step 2 - Form the ratio.

In this problem: Form the ratio: \(\frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4} = 0.75\).

\(\frac{3}{4} = 0.75\)

Step 3

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

In this problem: Apply the inverse function: \(\theta = \tan^{-1}(0.75) \approx 36.8699^{\circ}\).

\(\theta = \tan^{-1}(0.75) \approx 36.8699^{\circ}\)

Final Answer

\(\tan^{-1} \text{ with } \text{opposite} = 3,\; \text{adjacent} = 4 = \theta \approx 36.8699^{\circ}\)