Applications of the Sine Rule
Example
At a picnic by the river two children wondered how tall the cliff opposite the river was. By taking some measurements, like the angles and distances shown, can you help them find the height of the cliff?
Goal: Find length $AB$AB.
In triangle $ABC$ABC we only have angle measurements before we can use trigonometry to find length $AB$AB we will need a side length. Length $BC$BC can be found using triangle $BCD$BCD.
In triangle $BCD$BCD, angle B is $180-69-47=64$180−69−47=64° (angles in a triangle sum to $180$180)
Using the sine rule:
| $\frac{b}{\sin B}$bsinB | $=$= | $\frac{d}{\sin D}$dsinD | $d$d is length $BC$BC, named d because it is opposite angle $D$D |
| $\frac{143}{\sin64^\circ}$143sin64° | $=$= | $\frac{d}{\sin47^\circ}$dsin47° | then we can rearrange to find length $d$d |
| $\frac{143\sin47^\circ}{\sin64^\circ}$143sin47°sin64° | $=$= | $d$d | |
| $d$d | $=$= | $116.36$116.36 m |
Now we are only halfway through. We still need to find the height of the cliff.
Using tan we can complete the question
| $\tan39^\circ$tan39° | $=$= | $\frac{height}{116.36}$height116.36 |
| $height$height | $=$= | $116.36\times\tan39^\circ$116.36×tan39° |
| $height$height | $=$= | $94.2265$94.2265 ... |
So the cliff is approximately $94$94m tall.
Using the sine rule
The sine rule is useful when you want to find:
- a side length if you know 2 angles and another side
- an angle size if you know 1 angle and 2 corresponding side lengths
The best way to start problems involving applications of the sine rule is to label the angles of the triangle, label the corresponding sides and then use the most appropriate version of the sine rule.
$\frac{a}{\sin A}=\frac{b}{\sin B}$asinA=bsinB =$\frac{c}{\sin C}$csinC,
we can also use the alternative form
$\frac{\sin A}{a}=\frac{\sin B}{b}$sinAa=sinBb = $\frac{\sin C}{c}$sinCc
depending on what we are trying to use the law for.
The sine rule shows: that the lengths of the sides in a triangle are proportional to the
sines of the measures of the angles opposite them
Here are some worked examples.
Question 1
Question 2
Mae observes a tower at an angle of elevation of $12^\circ$12°. The tower is perpendicular to the ground.
Walking $67$67 m towards the tower, she finds that the angle of elevation increases to $35^\circ$35°.
Calculate the angle $\angle ADB$∠ADB.
Find the length of the side $a$a.
Round your answer to two decimal places.
Using the rounded value of the previous part, evaluate the height $h$h, of the tower.
Round your answer to one decimal place.
Question 3
Lucy travelled on a bearing of $39^\circ$39° from point A to point B. She then travelled on a bearing of $159^\circ$159° for $17$17 km to point C which is east of point A.
Find the size of the angle $\angle CAB$∠CAB.
Find the size of the angle $\angle ABC$∠ABC
Solve for $x$x, the distance Lucy would have to travel due east to return to point A.
Give your answer to two decimal places.