 # Sine Law

Lesson

If we have a right-angled triangle, we can use trigonometric ratios to relate the sides and angles: Here, $\sin A=\frac{a}{c}$sinA=ac and $\sin B=\frac{b}{c}$sinB=bc.

But what happens when we have a different kind of triangle? In a triangle like this, the same equations do not hold. We need to think of a different way to relate the sides and angles together.

## The sine rule

Let's start by drawing a line segment from the vertex $C$C perpendicular to the edge $c$c. We'll call the length of this segment $x$x. Since $x$x is perpendicular to $c$c, the two line segments meet at right angles. This means that we have divided our triangle into two right-angled triangles, and we can use the equations we already know. The relationships for the sines of the angles $A$A and $B$B is given by

$\sin A=\frac{x}{b}$sinA=xb and $\sin B=\frac{x}{a}$sinB=xa.

However, $x$x wasn't in our original triangle. So we want to find a relationship using only $A$A, $B$B, $a$a and $b$b. Multiplying the first equation by $b$b and the second by $a$a gives us

$x=b\sin A$x=bsinA and $x=a\sin B$x=asinB,

and equating these two equations eliminates the $x$x and leaves us with

$b\sin A=a\sin B$bsinA=asinB.

Dividing this last equation by the side lengths gives us the relationship we want:

$\frac{\sin A}{a}=\frac{\sin B}{b}$sinAa=sinBb.

We can repeat this process to find how these two angles relate to $c$c and $C$C, and this gives us the sine rule (sometimes called the law of sines).

The sine rule

For a triangle with sides $a$a, $b$b, and $c$c, with corresponding angles $A$A, $B$B, and $C$C,

$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$sinAa=sinBb=sinCc.

We can also take the reciprocal of each fraction to give the alternate form,

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$asinA=bsinB=csinC.

The sine rule shows that the lengths of the sides in a triangle are proportional to the sines of the angles opposite them.

## Finding a side length using the sine rule

Suppose we had the angles $A$A and $B$B and the length $b$b and we wanted to find the length $a$a. Using the form of the sine rule with numerator lengths $\frac{a}{\sin A}=\frac{b}{\sin B}$asinA=bsinB, we can make $a$a the subject by multiplying both sides by $\sin A$sinA. This gives

$a=\frac{b\sin A}{\sin B}$a=bsinAsinB.

#### Worked example

Solve: Find the length of $PQ$PQ to two decimal places. Think: The side we want to find is opposite a known angle, and we also know a matching side and angle. This means we can use the sine rule.

Do: $\frac{PQ}{\sin48^\circ}$PQsin48°​ $=$= $\frac{18.3}{\sin27^\circ}$18.3sin27°​ $PQ$PQ $=$= $\frac{18.3\sin48^\circ}{\sin27^\circ}$18.3sin48°sin27°​ $PQ$PQ $=$= $29.96$29.96 (to 2 d.p.)

## Finding an angle using the sine rule

Suppose we had the side lengths $a$a and $b$b, and the angle $B$B, and we want to find the angle $A$A. Using the form of the sine rule with numerator sines $\frac{\sin A}{a}=\frac{\sin B}{b}$sinAa=sinBb, we first multiply both sides by $a$a. This gives $\sin A=\frac{a\sin B}{b}$sinA=asinBb. We then take the inverse sine of both sides to make $A$A the subject, which gives

$A=\sin^{-1}\left(\frac{a\sin B}{b}\right)$A=sin1(asinBb).

#### Worked example

Solve: Find $\angle PRQ$PRQ to one decimal place. Think: The angle we want to find is opposite a known side, and we also know a matching angle and side. This means we can use the sine rule.

Do: $\frac{\sin R}{28}$sinR28​ $=$= $\frac{\sin39^\circ}{41}$sin39°41​ $\sin R$sinR $=$= $\frac{28\times\sin39^\circ}{41}$28×sin39°41​ $R$R $=$= $\sin^{-1}\left(\frac{28\times\sin39}{41}\right)$sin−1(28×sin3941​) $R$R $=$= $25.5^\circ$25.5° (to 1 d.p.)

#### Practice questions

##### Question 1

Find the value of the acute angle $x$x using the Sine Rule. ##### Question 2

Find the side length $a$a using the sine rule. ##### Question 3

Consider the triangle with two interior angles $C=72.53^\circ$C=72.53° and $B=31.69^\circ$B=31.69°, and one side length $a=5.816$a=5.816 metres. 1. Solve for the unknown interior angle $A$A.

2. Solve for $b$b.

3. Solve for $c$c.

### Outcomes

#### 10D.T3.01

Explore the development of the sine law within acute triangles

#### 10D.T3.03

Determine the measures of sides and angles in acute triangles, using the sine law and the cosine law