Trigonometry: Finding angles and sides

In a nutshell

$\sin()$ , $\cos()$ , $\tan()$ and their corresponding ratios can be used to find missing lengths and angles in a right-angled triangle.

Calculating a 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 and rearrange the formula to make the missing side the subject.
4.	Substitute in the values for the other side and the angle to get the length.

Example 1

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

Maths; Trigonometry; KS4 Year 10; Trigonometry: Finding angles and sides

Label the sides and angle:

Maths; Trigonometry; KS4 Year 10; Trigonometry: Finding angles and sides

$O$  and $H$ are the given sides. So, use SOH:

$\sin(50) = \frac{O}{H}$

So:

$O = H \times \sin(50)$

Substitute in the values:

$O = 10 \times \sin(50) = 7.660444....$

Round to 1 decimal place, like the question asks:

$\underline{O = 7.7m \ (1 \ d.p.)}$

Calculating an angle

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 the values for the sides.
5.	Perform the appropriate inverse trigonometric function to get the angle.

Example 2

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

Maths; Trigonometry; KS4 Year 10; Trigonometry: Finding angles and sides

First, label the diagram:

Maths; Trigonometry; KS4 Year 10; Trigonometry: Finding angles and sides

$O$ and $A$ are the important sides here. Hence, use TOA:

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

Substitute in the values:

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

Use the corresponding inverse trigonometric function:

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

$x= 22.61986....$ 

Round to 1 decimal place, like the question asks:

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