The Cosine Rule (VCE SSCE General Mathematics): Revision Notes

Worked Example: Finding an Unknown Side

Find side $c$, to two decimal places, in the triangle shown.

Step 1: Write down the given values and the required unknown value.

$a = 34, \quad b = 27, \quad c = ?, \quad C = 50°$

Step 2: We have two sides and the angle between them. To find side $c$, use:

$c^2 = a^2 + b^2 - 2ab \cos C$

Step 3: Substitute the given values:

$c^2 = 34^2 + 27^2 - 2 \times 34 \times 27 \times \cos 50°$

Step 4: Take the square root of both sides:

$c = \sqrt{34^2 + 27^2 - 2 \times 34 \times 27 \times \cos 50°}$

Step 5: Use your calculator to evaluate:

$c = 26.548...$

Step 6: Round to two decimal places:

The length of side $c$ is 26.55 units.

Worked Example: Finding an Unknown Angle

Find the largest angle, to one decimal place, in the triangle shown.

Step 1: Write down the given values.

$a = 6, \quad b = 4, \quad c = 5$

Step 2: The largest angle is always opposite the largest side. Since side $a = 6$ is the longest, we need to find angle $A$.

$A = ?$

Step 3: We have three sides. To find angle $A$, use:

$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

Step 4: Substitute the given values:

$\cos A = \frac{4^2 + 5^2 - 6^2}{2 \times 4 \times 5}$

Step 5: To find angle $A$, use the inverse cosine function:

$A = \cos^{-1}\left(\frac{4^2 + 5^2 - 6^2}{2 \times 4 \times 5}\right)$

Step 6: Use your calculator to evaluate. Make sure your calculator is in DEGREE mode.

Calculator tip: Put brackets around all the terms in the numerator (top), and also around all the terms in the denominator (bottom).

$A = 82.819...°$

Step 7: Round to one decimal place:

The largest angle is 82.8°.

Worked Example: Finding an Angle and a Bearing

A yacht left point $A$ and sailed $15$ km east to point $C$. Another yacht also started at point $A$ and sailed $10$ km to point $B$, as shown in the diagram. The distance between points $B$ and $C$ is $12$ km.

Part a: What was the angle between their directions as they left point $A$? Give the angle to two decimal places.

Step 1: Write the given values.

$a = 12, \quad b = 15, \quad c = 10$

Step 2: Write the form of the cosine rule for the required angle $A$:

$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

Step 3: Substitute the given values:

$\cos A = \frac{15^2 + 10^2 - 12^2}{2 \times 15 \times 10}$

Step 4: Find angle $A$:

$A = \cos^{-1}\left(\frac{15^2 + 10^2 - 12^2}{2 \times 15 \times 10}\right)$

Step 5: Use your calculator:

$A = 52.891°$

Step 6: Give the answer to two decimal places:

The angle was 52.89°.

Part b: Find the bearing of point $B$ from the starting point $A$, to the nearest degree.

Step 1: The bearing $\theta$ of point $B$ from starting point $A$ is measured clockwise from north.

Step 2: Consider the angles in the right angle at point $A$:

$\theta + 52.89° = 90°$

Step 3: Find the value of $\theta$:

$\theta = 90° - 52.89° = 37.11°$

Step 4: Write your answer as a three-figure bearing:

The bearing of point $B$ from point $A$ is 037°.

Worked Example: Bushwalker Problem

A bushwalker left his base camp and walked $10$ km in the direction $070°$. His friend also left the base camp but walked $8$ km in the direction $120°$.

Part a: Find the angle between their paths.

Step 1: Angles lying on a straight line add to $180°$:

$60° + A + 70° = 180°$

$A + 130° = 180°$

$A = 50°$

Step 2: The angle between their paths was 50°.

Part b: How far apart were they when they stopped walking? Give your answer to two decimal places.

Step 1: Write down the known values and the required unknown value.

$a = ?, \quad b = 8, \quad c = 10, \quad A = 50°$

Step 2: We have two sides and the angle between them. To find side $a$, use:

$a^2 = b^2 + c^2 - 2bc \cos A$

Step 3: Substitute in the known values:

$a^2 = 8^2 + 10^2 - 2 \times 8 \times 10 \times \cos 50°$

Step 4: Take the square root of both sides:

$a = \sqrt{8^2 + 10^2 - 2 \times 8 \times 10 \times \cos 50°}$

Step 5: Use a calculator:

$a = 7.820...$

Step 6: Round to two decimal places:

Distance between them was 7.82 km.