If $A,B,C$A,B,C are the vertices of a triangle and $a,b,c$a,b,c are the side lengths opposite to each angle respectively, then the sine rule is given by:
$\frac{\sin A}{a}$sinAa$=$=$\frac{\sin B}{b}$sinBb$=$=$\frac{\sin C}{c}$sinCc
Looking at part of this sine rule:
$\frac{\sin A}{a}=\frac{\sin B}{b}$sinAa=sinBb
If three of the four values are known, then we can find the missing angle or side.The important point to note is that we need to be able to relate angles to their opposite sides.
We are now going to use this rule to find these unknown quantities in real world contexts. A good way to begin any question involving a triangle is to label the angles and their corresponding sides using the letters from the formula above. After that is done correctly we can use the version of the sine rule that is most convenient.
Exploration
There are three flight paths joining Adelaide, Sydney, and Brisbane to one another. The flight path joining Adelaide and Sydney is $1601$1601 km long, and the flight path joining Sydney and Brisbane is $732$732 km long. The angle between the two flight paths meeting at Adelaide is $20^\circ$20°.
An aircraft will travel from Sydney to Adelaide via Brisbane. Find $\theta$θ, the angle between the two flight paths meeting at Brisbane, to the nearest degree.
The line segments between each city forms a triangle. We know the lengths of two sides and one angle, so we can use the sine rule ($\frac{\sin A}{a}=\frac{\sin B}{b}$sinAa=sinBb) to find the size of the angle $\theta$θ. Here is the triangle without the underlying map, with sides and angles labelled in a corresponding way:
| $\frac{\sin A}{a}$sinAa | $=$= | $\frac{\sin B}{b}$sinBb | Write down the sine rule. |
| $\frac{\sin\theta}{1160}$sinθ1160 | $=$= | $\frac{\sin20^\circ}{732}$sin20°732 | Substitute in the known values. |
| $\sin\theta$sinθ | $=$= | $\frac{1160\sin20^\circ}{732}$1160sin20°732 | Multiplying both sides by $1160$1160. |
| $\theta$θ | $=$= | $\sin^{-1}\left(\frac{1160\sin20^\circ}{732}\right)$sin−1(1160sin20°732) | Take the inverse sine of both sides. |
| $\theta$θ | $=$= | $33^\circ$33° | Calculate the answer, rounded to nearest degree. |
Practice questions
Question 1
Mae observes a tower at an angle of elevation of $12$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$35° .
Calculate the size of $\angle ADB$∠ADB.
Find the value of $a$a, correct to two decimal places.
Using the rounded value of the previous part, find $h$h where $h$h m the height of the tower.
Give your answer correct to one decimal place.
Question 2
Judy and Xavier need to build a bridge over a river, along the line $ST$ST. In order to find out how long the bridge needs to be, Judy stays at point $T$T and measures the angle at $T$T to point $S$S to be $112^\circ$112° while Xavier walks for $355$355 m down the river to point $V$V where the angle to point $S$S is $28^\circ51'$28°51′.
If the length of the bridge they will build is $d$d m, find $d$d correct to one decimal place.
Question 3
The stage director of a musical wants a spotlight shining on a $5.3$5.3 m section of the stage for a particular song. She determines that the further edge of the light needs to hit the stage at an angle of $29^\circ$29° so that certain props are hidden. The spotlight is set to spread light at an angle of $23^\circ$23°, but its height can be adjusted.
First find the value of $x$x, correct to two decimal places.
Hence find the height, $h$h m, that the stage director must raise the spotlight to for this song, correct to $2$2 decimal places.