Using the Sine or Cosine Rule
• Introduction

This worksheet is about using the sine and the cosine rule to find the lengths of sides or the sizes of angles.

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.

We can use the cosine rule to find the length of a side when we know two sides of a triangle and the angle between these two sides.

Example 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 ∠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°

Example 2

In  triangle PQR, which is not drawn to scale, PQ = 7 cm, QR = 22 cm, and ∠PQR = 145º.

Find the length PR to 3 sig figs.

Using the cosine rule,

PR2 = 72 + 222 - 2 x 7 x 22 x cos145º

= 49 + 484 - 308 x (-0.819)

= 785.3

PR = √785.3 ≈ 28.0 cm

The sine rule states that:

 a = b = c sin A sin B sin C

We can use the sine rule to find an angle in the triangle when we know one other angle and the length of two sides.

Remember that, if the known angle is between the two given sides, we use the cosine rule.

We can use the sine rule to find the length of a side when we know two angles and an opposite side.

Example 3

In  triangle PQR, which is not drawn to scale,

QR = 10 cm, PR = 12 cm and ∠PQR = 95°.

Use the sine rule to find ∠PRQ to 1 dp.

First we must find ∠QPR using the sine rule.

Using the sine rule,

 12 = 10 sin 95° sin ∠QPR

 sin ∠QPR = 10 x sin 95° 12

sin ∠QPR = 0.83016...

∠QPR = sin-1 (0.83016) = 56.1°

∠PQR = 180° - 56.1° - 95° = 28.9°

Example 4

In  triangle PQR, which is not drawn to scale,

PQ = 7 cm, ∠QPR = 72° and ∠PRQ = 24°.

Use the sine rule to find the length of PR in cm to 1 dp.

First we must find ∠PQR = 180° - 72° - 24° = 84°.

Using the sine rule,

 PR = 7 sin 84° sin 24°

 PR = 7 x sin 84° sin 24°

PR = 7 x sin 84° ÷ sin 24°

PR = 17.1 cm

• Question 1

In  triangle PQR, which is not drawn to scale,

PQ = 8 cm, QR = 10 cm, and ∠PQR = 137º.

Find the length PR in cm to 3 sig figs.

(just write the number)

• Question 2

In  triangle PQR, which is not drawn to scale,

PQ = 8 cm, ∠QPR = 33° and ∠PRQ = 40°

Find the length of QR in cm to 1 dp.

(Just write the number)

• Question 3

In  triangle PQR, which is not drawn to scale,

PQ = 5 cm, QR = 11 cm, and PR = 12 cm.

Find ∠QPR to the nearest degree.

(Just write the number)

• Question 4

In  triangle PQR, which is not drawn to scale,

PQ = 8 cm, QR = 10 cm and ∠QPR = 33°

Find ∠QRP to 1 dp.

(Just write the number)

• Question 5

In  triangle PQR, which is not drawn to scale,

PQ = 13 cm, PR = 10 cm and ∠PRQ = 131°

Find ∠QPR to 1 dp.

(Just write the number)

• Question 6

In  triangle PQR, which is not drawn to scale,

PQ = 5 cm, QR = 11 cm, and ∠PQR = 95º.

Find the length PR to 3 sig figs.

(just write the number)

• Question 7

In  triangle PQR, which is not drawn to scale,

PQ = 13 cm, ∠QPR = 11° and ∠PRQ = 131°

Find the length of QR in cm to 1 dp.

(Just write the number)

• Question 8

In  triangle PQR, which is not drawn to scale,

PQ = 23 cm, PR = 18 cm and ∠PRQ = 131°

Find ∠QPR to 1 dp.

(Just write the number)

• Question 9

In  triangle PQR, which is not drawn to scale,

PQ = 8 cm, QR = 10 cm, and PR = 16 cm.

Find ∠PQR to the nearest degree.

(Just write the number)

• Question 10

In  triangle PQR, which is not drawn to scale,

PQ = 22 cm, QR = 16 cm and ∠QPR = 22°

Find ∠QRP to 1 dp.

(Just write the number)

Progress