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 Opposite and the Hypotenuse.

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

This reminds us of the equation:

Sin θ = Opposite / Hypotenuse

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 = Sin−1 (Opposite / Hypotenuse)

Slider

The slider below gives another example of finding the angle of a right triangle (if the hypotenuse and opposite are known).

Open the slider in a new tab

See Also

What is an angle? What is a right triangle? What is the opposite? What is the hypotenuse?