The sine rule - Higher

The angles are labelled with capital letters. The opposite sides are labelled with lower case letters. Notice that an angle and its opposite side are the same letter.

Triangle (ABC) and lengths a, b and c

The sine rule is: \frac{a}{\sin{A}} = \frac{b}{\sin{B}} = \frac{c}{\sin{C}}

This version is used to calculate lengths.

It can be rearranged to: \frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c}

This version is used to calculate angles.

Example

Calculate the angle PRQ. Give the answer to three significant figures.

Triangle (PQR) with lengths 4cm and 9cm. Angles x and 75 degrees

Use the form \frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c} to calculate the angle.

\frac{\sin{x}}{4} = \frac{\sin{75}}{9}

\sin{x} = \frac{4 \times \sin{75}}{9}

\sin{x} = 0.429300 \dotsc. Do not round this answer yet.

To calculate the angle use the inverse sin button on the calculator ( \sin{x}^{-1}).

x = 25.4^\circ

Question

Calculate the length QR. Give the answer to three significant figures.

Triangle (PQR) with length 7.1cm. Angles x, 105 and 53 degrees

Use the form \frac{a}{\sin{A}} = \frac{b}{\sin{B}} = \frac{c}{\sin{C}} to calculate the length.

\frac{\text{QR}}{\sin{53}} = \frac{7.1}{\sin{105}}

\text{QR} = \frac{7.1 \times \sin{53}}{\sin{105}}

QR = 5.87 cm

Using the sine curve to calculate obtuse angles

Question

Calculate angle ACB

Triangle (ABC) with angle 32 degrees

Use the form \frac{\sin{A}}{a} = \frac{\sin{B}}{b} = \frac{\sin{C}}{c} to calculate the angle.

\frac{\sin{C}}{5.7} = \frac{\sin{32}}{3.1}

\sin{C} = \frac{5.7 \times \sin{32}}{3.1}

\sin{C} = 0.974367 \dotsc

C = 77°

Angle ACB is obtuse therefore angle C cannot be 77°.

Use the sine curve to calculate the obtuse angle.

Sine curve

\sin{77} = \sin{(180 - 77)}

C must be 103°.