The cosine rule - Higher

The cosine rule is: a^2 = b^2 + c^2 - 2bc \cos{A}

This version is used to calculate lengths.

It can be rearranged to: \cos{A} = \frac{b^2 + c^2 - a^2}{2bc}

This version is used to calculate angles.

Example

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

Triangle (ABC) with sides 3cm and 7cm and angle 35degrees

Use the form a^2 = b^2 + c^2 - 2bc \cos{A} to calculate the length.

\text{BC}^2 = 3^2 + 7^2 - 2 \times 3 \times 7 \cos{35}

\text{BC}^2 = 23.59561414 \dotsc. Do not round this answer yet.

BC = 4.86 cm

Question

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

Triangle (PQR) with lengths 4cm, 4.2cm and 6.9cm. Unknown angle y

Use the form \cos{A} = \frac{b^2 + c^2 - a^2}{2bc} to calculate the angle.

\cos{y} = \frac{4^2 + 6.9^2 - 4.2^2}{2 \times 4 \times 6.9}

\cos{y} = 0.8327898 \dotsc. Do not round this answer yet.

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

y = 33.6°