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7.03 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

Given \triangle ABC consists of angles A, B and C which appear opposite sides a, b and c respectively:

a
Write an expression for c^2 using the cosine rule.
b
Evaluate \cos 90\degree.
c
If \triangle ABC is a right-angled triangle with \angle C=90\degree, what does the expression given in part (a) simplify to?
5

Consider the following triangle:

a

Find an expression for a^{2} by using Pythagoras' theorem in \triangle BCD.

b

Find an expression for h^{2} by using Pythagoras' theorem in \triangle ACD.

c

Find an expression for x in terms of \cos A.

d

Substitute your expressions for h^{2} and x into your expression for a^{2} to prove the cosine rule.

Unknown sides
6

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

7

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
k
l
8

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

Find the exact length of side AB in centimetres.

9

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

Unknown angles
10

In \triangle QUV, v = 6, q = 10 and u = 12. Find the value of \cos U.

11

For each of the following triangles, find the value of the pronumeral in degrees. Round your answers to two decimal places.

a
b
c
d
e
f
g
h
12

For each of the following triangles, find \theta to the nearest degree:

a
b
13

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

14

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

15

A teacher is writing exam questions for her maths class. She draws a triangle, labels the vertices A, B and C and labels the opposite sides a = 5, b = 8 and c = 15 respectively.

She wants to ask students to find the size of \angle A. Explain why there is an error with her question.

16

A triangle has sides of length 13 \text{ cm}, 15 \text{ cm} and 5 \text{ cm}. Find the value of x, the largest angle in the triangle to the nearest degree.

17

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

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Outcomes

4.2.1.3

establish and use the sine (ambiguous case is required) and cosine rules and the formula 𝐴rea=1/2 bc sin𝐴 for the area of a triangle

4.2.1.4

construct mathematical models using the sine and cosine rules in two- and three-dimensional contexts (including bearings in two-dimensional context) and use the model to solve problems; verify and evaluate the usefulness of the model using qualitative statements and quantitative analysis

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