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

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Example 1

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

Label the sides and angle:

$$O$$​ 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 $$x$$​.

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

Example 2

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

First, label the diagram:

$$O$$​ 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: ​

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