# 7.06 The cosine rule

Worksheet
Cosine rule
1

Write an expression for \cos \theta using the cosine rule for the following triangle:

2

Consider the triangle given below:

a
Write an expression for \cos A using the cosine rule.
b
Write an expression for a^2 using the cosine rule.
c
Write an expression for b using the cosine rule.
3

Consider the triangle given below:

a
Write an expression for \cos Q using the cosine rule.
b
Write an expression for r^2 using the cosine rule.
4

To use the cosine rule to find the length ofAC, which angle would need to be given?

Unknown sides
5

Find the length of the missing side in each of the following triangles using the cosine rule. Round your answers to two decimal places.

a
b
c
d
e
f
g
h
i
j
6

In \triangle ABC, \cos C = \dfrac{8}{9}:

Find the exact length of side AB in centimetres.

7

In \triangle QUV, q = 5, u = 6 and \cos V = \dfrac{3}{5}. Find the value of v.

Unknown angles
8

For each of the following triangles, find the value of the pronumeral to the nearest degree:

a
b
c
d
e
f
g
h
9

Find the value of \theta in the following triangle. Round your answer to the nearest hundredth of a degree.

10

Find the value of B in the following triangle. Round your answer to the nearest second.

11

In \triangle QUV, v = 8, u = 9 and q = 15. Solve for \cos Q.

12

The sides of a triangle are in the ratio 4:5:8. Find \theta, the smallest angle in the triangle, to the nearest degree.

13

A triangle has sides of length 11 \text{ cm}, 18 \text{ cm} and 8 \text{ cm}. Find x, the largest angle in the triangle, to the nearest degree.

Applications
14

Mae went for a bike ride on Sunday morning from Point A to Point B, which was 18 \text{ km} long. She then took a 126 \degree turn and rode from Point B to Point C, which was 21 \text{ km} long.

Find x, the distance in kilometres from her starting point to Point C to two decimal places.

15

A goal has posts that are 2 \text{ m} apart. Buzz shoots for the goal when he is 2.6 \text{ m} from one post and 3.1 \text{ m} from the other post.

Find the size of the angle, x, in which he can score a goal. Round your answer to the nearest degree.

16

Find the length of the diagonal, x, in parallelogram ABCD.

17

Consider the parallelogram in the given diagram that has a side of length 13 \text{ cm} and a diagonal of length 58 \text{ cm}:

Find the value of x. Round your answer to one decimal place.

18

In a sailing boat race, teams must start at buoy A and navigate around buoys B and C before returning to buoy A to cross the line. The first leg of the race is 100.6 \text{ km} long, the second leg of the race is 190.3 \text{ km} long, and the angle between these legs is 143 \degree.

a

Find x, the distance of the third leg of the race, correct to one decimal place.

b

Hence, find the total length of the race, correct to one decimal place.

19

A bridge connects two towns on either side of a gorge, where one side of the gorge is inclined at 59 \degree and the other side is inclined at 70 \degree. The length of the steeper incline is 59.1 \text{ m}.

Find x, the length of the bridge. Round your answer correct to one decimal place.