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Using the Sine Function to Find the Angle

(KS3, Year 8)

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:

theta equals sine function

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:

sine opposite hypotenuse image

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? right triangle o 2 h 4 theta

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.

right triangle o known h known

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)

Lesson Slides

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

Interactive Widget

Here is an interactive widget to help you learn about the sine function on a right triangle.

What Is the Inverse Sine Function?

The inverse sine function is the opposite of the sine function. The sine function takes in an angle, and gives the ratio of the opposite to the hypotenuse:

theta sin oh

The inverse sine function, sin⁻¹, goes the other way. It takes the ratio of the opposite to the hypotenuse, and gives the angle:

oh inverse sin theta

Switch Sides, Invert the Sine

You may see the sine function in an equation:

To make θ the subject of the equation, take the inverse sine of both sides. The inverse sine cancels out the sine on the left hand side of the equals side, so the equation looks as below:

Comparing the two equations, the sine has moved from one side of the equals sign to the other and has changed from sin to sin⁻¹.

(Note: the reverse is also true. A sin⁻¹ can be moved to the other side of the equals sign, where it becomes a sin.)