 # Finding Angles Using the Cosine Rule

In this worksheet, students find an angle given three sides of a triangle by using the cosine rule. Key stage:  KS 4

Curriculum topic:  Geometry and Measures

Curriculum subtopic:  Know the Exact Values of Sin, Cos and Tan

Difficulty level:   ### QUESTION 1 of 10

This worksheet is about finding angles using the cosine rule.

Look at  triangle ABC below The cosine rule states that:

a2 = b2 + c2 - 2bc cosA

We can use the cosine rule to find an angle when we know all three sides of a triangle.

The side of length a will be opposite the angle we are trying to find.

Example:

In  triangle PQR, which is not drawn to scale,

PQ = 9 cm, QR = 10 cm, and PR = 15 cm.

Use the cosine rule to find ∠PQR to the nearest degree. Using the cosine rule,

152 = 92 + 102 - 2 x 9 x 10 x cosθº

225 = 81 + 100 - 180cosθº

180cosθº = 81 + 100 - 225 = -44

cosθº = -44 ÷ 180 = -0.24444444....

θº = cos-1(-0.24444444...) = 104°

In  triangle PQR, which is not drawn to scale,

PQ = 9 cm, QR = 10 cm, and PR = 15 cm.

Use the cosine rule to find ∠QPR to the nearest degree.

(Just write the number) In  triangle PQR, which is not drawn to scale,

PQ = 9 cm, QR = 10 cm, and PR = 15 cm.

Use the cosine rule to find ∠QRP to the nearest degree.

(Just write the number) In  triangle PQR, which is not drawn to scale,

PQ = 4.5 cm, QR = 10 cm, and PR = 12 cm.

Calculate ∠PQR to the nearest degree.

(Just write the number) In  triangle PQR, which is not drawn to scale,

PQ = 4.5 cm, QR = 10 cm, and PR = 12 cm.

Use the cosine rule to find ∠PRQ to the nearest degree.

(Just write the number) In  triangle PQR, which is not drawn to scale,

PQ = 4.5 cm, QR = 10 cm, and PR = 12 cm.

Use the cosine rule to find ∠PRQ to the nearest degree.

(Just write the number) In  triangle ABC, which is not drawn to scale, AB = 10 cm, BC = 6 cm, and AC = 12 cm.

Use the cosine rule to find ∠ABC to the nearest degree.

(Just write the number)

In  triangle ABC, which is not drawn to scale, AB = 10 cm, BC = 6 cm, and AC = 12 cm.

Use the cosine rule to find ∠BAC to the nearest degree.

(Just write the number)

In  triangle ABC, which is not drawn to scale, AB = 10 cm, BC = 6 cm, and AC = 8 cm.

Use the cosine rule to find ∠ACB to the nearest degree.

(Just write the number)

In  triangle ABC, which is not drawn to scale, AB = 10 cm, BC = 10 cm, and AC = 10 cm.

Use the cosine rule to find ∠ACB to the nearest degree.

(Just write the number)

In  triangle ABC, which is not drawn to scale, a = 10 cm, b = 5.5 cm, and c = 12 cm.

Use the cosine rule to find ∠A to the nearest degree.

(Just write the number)

• Question 1

In  triangle PQR, which is not drawn to scale,

PQ = 9 cm, QR = 10 cm, and PR = 15 cm.

Use the cosine rule to find ∠QPR to the nearest degree.

(Just write the number) 40
EDDIE SAYS
10² = 9² + 15² - 2 × 9 × 15 × cosθ°
• Question 2

In  triangle PQR, which is not drawn to scale,

PQ = 9 cm, QR = 10 cm, and PR = 15 cm.

Use the cosine rule to find ∠QRP to the nearest degree.

(Just write the number) 36
EDDIE SAYS
9² = 10² + 15² - 2 × 10 × 15 × cosθ°
• Question 3

In  triangle PQR, which is not drawn to scale,

PQ = 4.5 cm, QR = 10 cm, and PR = 12 cm.

Calculate ∠PQR to the nearest degree.

(Just write the number) 22
EDDIE SAYS
12² = 10² + 4.5² - 2 × 10 × 4.5 × cosθ°
• Question 4

In  triangle PQR, which is not drawn to scale,

PQ = 4.5 cm, QR = 10 cm, and PR = 12 cm.

Use the cosine rule to find ∠PRQ to the nearest degree.

(Just write the number) 21
EDDIE SAYS
4.5² = 10² + 12² - 2 × 10 × 12 × cosθ°
• Question 5

In  triangle PQR, which is not drawn to scale,

PQ = 4.5 cm, QR = 10 cm, and PR = 12 cm.

Use the cosine rule to find ∠PRQ to the nearest degree.

(Just write the number) 53
EDDIE SAYS
10² = 4.5² + 12² - 2 × 4.5 × 12 × cosθ°
• Question 6

In  triangle ABC, which is not drawn to scale, AB = 10 cm, BC = 6 cm, and AC = 12 cm.

Use the cosine rule to find ∠ABC to the nearest degree.

(Just write the number)

94
EDDIE SAYS
12² = 10² + 6² - 2 × 10 × 6 × cosθ°
• Question 7

In  triangle ABC, which is not drawn to scale, AB = 10 cm, BC = 6 cm, and AC = 12 cm.

Use the cosine rule to find ∠BAC to the nearest degree.

(Just write the number)

30
EDDIE SAYS
6² = 10² + 12² - 2 × 10 × 12 × cosθ°
• Question 8

In  triangle ABC, which is not drawn to scale, AB = 10 cm, BC = 6 cm, and AC = 8 cm.

Use the cosine rule to find ∠ACB to the nearest degree.

(Just write the number)

90
EDDIE SAYS
10² = 6² + 8² - 2 × 6 × 8 × cosθ°
This is a 6, 8, 10 Pythagorean Triad.
• Question 9

In  triangle ABC, which is not drawn to scale, AB = 10 cm, BC = 10 cm, and AC = 10 cm.

Use the cosine rule to find ∠ACB to the nearest degree.

(Just write the number)

60
EDDIE SAYS
10² = 10² + 10² - 2 × 10 × 10 × cosθ°
This is an equilateral triangle.
• Question 10

In  triangle ABC, which is not drawn to scale, a = 10 cm, b = 5.5 cm, and c = 12 cm.

Use the cosine rule to find ∠A to the nearest degree.

(Just write the number)