Right-Angled Trigonometry

Syllabus

review and use the trigonometric ratios to find the length of an unknown side or the size of an unknown angle in a right-angled triangle

Trigonometry is used to calculate unknown side lengths and angles of triangles.

Before you can perform any trigonometric calculations you need to be able to identify the sides of a right-angled triangle. The sides are labeled relative to the position of the given angle as shown below:

Important

The trigonometric ratios are as follows:

• $\sin A=\dfrac{\text{Opposite (opp)}}{\text{Hypotenuse (hyp)}}$

• $\cos A=\dfrac{\text{Adjacent (adj)}}{\text{Hypotenuse (hyp)}}$

• $\tan A=\dfrac{\text{Opposite (opp)}}{\text{Adjacent (adj)}}$

note

These are available on your reference sheet

Finding Side Lengths

To find an unknown side length using right-angled trigonometry you need to know at least one acute angle and one side length. Using the known variables and the unknown side, choose the appropriate ratio and substitute in your values. Solve the equation to find the unknown length.

note

If you know two sides of a right-angled triangle you can use Pythagoras' theorem ($a^2+b^2=c^2$, where $c$ is the hypotenuse) to find the last side.

Example 1

Find the value of $x$ rounded to 3 significant figures.

Solution

$$\begin{aligned} \tan\theta&=\frac{\text{opp}}{\text{adj}}&&\text{(Choose the appropriate ratio)}\\ \tan37^\circ&=\frac{x}{11}&&\text{(Substitute)}\\ 11\times\tan37^\circ&=x&&\text{(Multiply both sides by 11)}\\ x&=8.289...\\ x&\approx8.29\text{ mm}&&\text{(Rounded to appropraite number}\\ &&&\text{of s.f. and include units)} \end{aligned}$$

Example 2

Find the value of $n$ rounded to 2 decimal places.

Solution

$$\begin{aligned} \sin\theta&=\frac{\text{opp}}{\text{hyp}}&&\text{(Choose the appropriate ratio)}\\ \sin28^\circ&=\frac{3}{n}&&\text{(Substitute)}\\ n&=\frac{3}{\sin28^\circ}&&\text{(Multiply both sides by $n$ and}\\ &&&\text{divide by $\sin28^\circ$)}\\ n&=6.390...\\ n&\approx6.39&&\text{(Rounded to appropraite number of d.p.)} \end{aligned}$$

note

In the diagram, $n$ already has units so you do not need to include them.

Finding Angles

To find an unknown angle using right-angled trigonometry you need to know at least two side lengths. Using the known sides, choose the appropriate ratio and substitute in your values. Solve the equation to find the unknown angle.

note

If you know two angles in a triangle you can use angle sum ($180^\circ$) to find the last angle.

Example 3

Find the value of $\theta$ correct to 3 s.f.

Solution

$$\begin{aligned} \sin\theta&=\frac{\text{opp}}{\text{hyp}}&&\text{(Choose the appropriate ratio)}\\ \sin\theta&=\frac{3}{7}&&\text{(Substitute)}\\ \theta&=\sin^{-1}\left(\frac{3}{7}\right)&&{\text{\htmlClass{casio}{qj3P7)=}}}\\ \theta&=25.3769...\\ \theta&\approx25.4^\circ&&\text{(Rounded to appropraite number}\\ &&&\text{of s.f. and include units)} \end{aligned}$$

Example 4

Find the value of $\phi$ correct to the nearest minute.

Solution

Important

Always ensure your units are consistent.

$$\begin{aligned} \cos\phi&=\frac{\text{adj}}{\text{hyp}}&&\text{(Choose the appropriate ratio)}\\ \cos\phi&=\frac{23\text{ mm}}{70\text{ mm}}&&\text{(Substitute and ensure consistent units)}\\ \phi&=\cos^{-1}\left(\frac{23}{70}\right)&&{\text{\htmlClass{casio}{qk23P70)=}}}\\ \phi&=70.8179...\\ \phi&=70^\circ\,49'\,4.476...''&&{\text{\htmlClass{casio}{x}}}\\ \phi&\approx70^\circ\,49'&&\text{(Rounded nearest minute)} \end{aligned}$$

note

Minutes and seconds are out of 60. This means they round up from 30.

Had the answer been $70^\circ\,49'\,34.476...$ then it would have rounded up to $70^\circ\,50'$.