3.09 The cosine rule

For each of the given triangles, determine if there is enough information to find all the remaining sides and angles in the triangle using only the cosine rule:

a

Three sides are known:

b

Two of the angles and the side included between them are known:

c

Two of the angles and a side not included between them are known:

d

Two of the sides and an angle included between them are known:

For each of the following triangles, find the value of the pronumeral using the cosine rule. Round your answers to two decimal places.

a

b

c

d

e

f

g

h

i

Triangle QUV has side lengths v = 7, u = 10 and an angle at Q such that \cos Q = \dfrac{1}{5}. Find the value of q, where q is the length of the side opposite angle Q.

Triangle QUV has side lengths q = 63, u = 36 and \angle V = 116 \degree. Find the value of v, where v is the length of the side opposite \angle V. Round your answer to two decimal places.

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

Triangle QUV has side lengths q = 5, u = 10 and v = 12. Find the size of \angle V, where V is the angle opposite the side of length v. Round your answer to the nearest minute.

Triangle QUV has side lengths v = 23.5, u = 15.6 and q = 10.0. Find the size of \angle Q, where Q is the angle opposite the side of length q. Round your answer to the nearest minute.

The sides of a triangle are in the ratio 4:7:8. Find the size of x, the smallest angle in the triangle. Round your answer to the nearest minute.

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 what the problem is with her question.

Use the cosine rule to prove that any angle in a triangle with three equal sides is equal to 60 \degree.