Finding missing angles in a triangle

In a nutshell

Calculating the missing angle of a right-angled triangle is very similar to calculating the missing length. However, it is important to first learn about inverse trigonometric functions.

The inverse trigonometric functions

Inverse trigonometric functions invert the trigonometric function, giving the value of the angle $x$ .

Function	What it's called	examples
$\sin^{-1}()$	Inverse sine	If $\sin(x) = 0.5$ , then $x = \sin^{-1}(0.5)$
$\cos^{-1}()$	Inverse cosine	If $\cos(x) = 0.8$ , then $x = \cos^{-1}(0.8)$
$\tan^{-1}()$	Inverse tangent	If $\tan(x) = 0.42$ , then $x = \tan^{-1}(0.42)$

Note: The inverse function $\sin^{-1}()$ is NOT the same as $\frac{1}{\sin()}$ ! It is just notation to tell us that it is the inverse of the $\sin()$  function. The same goes for the other two trigonometric functions.

Calculating the missing angle

The process of finding the missing angle of a right angled triangle is very similar to finding the missing side.

PROCEDURE

1.	Label the sides and angle of the triangle.
2.	Work out what trigonometric ratio you have to use (SOH, CAH or TOA).
3.	Write down the corresponding formula.
4.	Substitute in your values for the sides.
5.	Perform the appropriate inverse trigonometric function to get the angle.

Example

Find the value of $x$ in the diagram to 1 decimal place.

Maths; Trigonometry; KS3 Year 7; Finding missing angles in a triangle

Label the triangle:

Maths; Trigonometry; KS3 Year 7; Finding missing angles in a triangle

You have the opposite and adjacent here. So, you have O and A. Hence, use TOA:

$\tan(x) = \frac{opposite}{adjacent}=\frac{O}{A}$

Substitute in your values:

$\tan(x)=\frac{5}{12}$

Perform the inverse tangent function to get $x$ by itself:

If $\tan(x) = \frac{5}{12}$ then $x = \tan^{-1}(\frac{5}{12})$

$x=22.619864....$

$\underline{x=22.6 \degree \ (1 d.p.)}$