# Using the Sine Function to Find the Angle

## Using the Sine Function to Find the Angle of a Right Triangle

The sine function relates a given angle to the opposite side and hypotenuse of a right triangle.

The angle (labelled θ) is given by the formula below:

In this formula, **θ** is an angle of a right triangle, the opposite is the length of the side opposite the angle and the hypotenuse is the length of longest side. sin^{−1} is the inverse sine function (see **Note**). The image below shows what we mean:

## How to Use the Sine Function to Find the Angle of a Right Triangle

Finding the angle of a right triangle is easy when we know the opposite and the hypotenuse.

### Question

What is the angle of the right triangle shown below?

### Step-by-Step:

# 1

Start with the formula:

θ = sin^{−1} (opposite / hypotenuse)

**Don't forget:** sin^{−1} is the inverse sine function (it applies to everything in the brackets) **and** / means ÷

# 2

Substitute the length of the opposite and the length of the hypotenuse into the formula. In our example, the opposite is 2 cm and the hypotenuse is 4 cm.

θ = sin^{−1} (2 / 4)

θ = sin^{−1} (2 ÷ 4)

θ = sin^{−1} (0.5)

θ = 30°

### Answer:

The angle of a right triangle with an opposite of 2 cm and a hypotenuse of 4 cm is 30°.

## Remembering the Formula

Often, the hardest part of finding the unknown angle is remembering which formula to use.

Whenever you have a right triangle where you know two sides and have to find an unknown angle...

......think trigonometry...

...............think sine, cosine or tangent...

........................think **SOH CAH TOA**.

Looking at the example above, we know the **O**pposite and the **H**ypotenuse.

The two letters we are looking for are **OH**, which comes in the **SOH** in **SOH** CAH TOA.

This reminds us of the equation:

**S**in θ = **O**pposite / **H**ypotenuse

We want the angle, θ, not the sine of the angle, sin θ. To do this, we need to taken the inverse sine, sin^{−1} (see **Note**).

θ = **S**in^{−1} (**O**pposite / **H**ypotenuse)