7.03 The cosine rule

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.

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.

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

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.

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

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

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

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

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

a

b

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.

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.

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