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.
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.
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.
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
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.
\(\sin{77} = \sin{(180 - 77)}\)
C must be 103°.