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Finding an Angle Using the Sine Rule

In this worksheet, students find an angle given one angle and two sides in a triangle by using the sine rule.

'Finding an Angle Using the Sine Rule' worksheet

Key stage:  KS 4

Curriculum topic:  Geometry and Measures

Curriculum subtopic:  Apply Pythagoras' Theorem and Trigonometric Ratios to Find Angles and Lengths in 2D Figures

Difficulty level:  

down

Worksheet Overview

QUESTION 1 of 10

This worksheet is about finding the length of a side using the sine rule.

Look at  triangle ABC below

 

 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.

 

Example:

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.

 

Answer:

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°

In  triangle PQR, which is not drawn to scale,

 

QR = 8 cm, PR = 10 cm and ∠PQR = 97°

 

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

 

(Just write the number)

 

In  triangle PQR, which is not drawn to scale,

 

PQ = 5 cm, PR = 10 cm and ∠PQR = 93°

 

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

 

(Just write the number)

 

In  triangle PQR, which is not drawn to scale,

 

PQ = 9 cm, PR = 15 cm and ∠QRP = 37°

 

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

 

(Just write the number)

 

In  triangle PQR, which is not drawn to scale,

 

PQ = 9 cm, QR = 10 cm and ∠QRP = 37°

 

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

 

(Just write the number)

 

In  triangle PQR, which is not drawn to scale,

 

PQ = 9 cm, QR = 10 cm and ∠QPR = 41°

 

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

 

(Just write the number)

 

In  triangle PQR, which is not drawn to scale,

 

PQ = 14 cm, QR = 10 cm and ∠QPR = 41°

 

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

 

(Just write the number)

 

In  triangle PQR, which is not drawn to scale,

 

PQ = 14 cm, PR = 10 cm and ∠PRQ = 123°

 

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

 

(Just write the number)

 

In  triangle PQR, which is not drawn to scale,

 

QR = 7.5 cm, PR = 10 cm and ∠PQR = 40°

 

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

 

(Just write the number)

 

In  triangle PQR, which is not drawn to scale,

 

PQ = 8 cm, PR = 11 cm and ∠PQR = 111°

 

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

 

(Just write the number)

 

In  triangle PQR, which is not drawn to scale,

 

PQ = 6 cm, PR = 12 cm and ∠QRP = 30°

 

Use the sine rule to find ∠PQR.

 

(Just write the number)

 

  • Question 1

In  triangle PQR, which is not drawn to scale,

 

QR = 8 cm, PR = 10 cm and ∠PQR = 97°

 

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

 

(Just write the number)

 

CORRECT ANSWER
52.6
EDDIE SAYS
10 = 8
sin 97° sin ∠QPR
  • Question 2

In  triangle PQR, which is not drawn to scale,

 

PQ = 5 cm, PR = 10 cm and ∠PQR = 93°

 

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

 

(Just write the number)

 

CORRECT ANSWER
30.0
EDDIE SAYS
10 = 5
sin 93° sin ∠QRP
  • Question 3

In  triangle PQR, which is not drawn to scale,

 

PQ = 9 cm, PR = 15 cm and ∠QRP = 37°

 

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

 

(Just write the number)

 

CORRECT ANSWER
60.4
EDDIE SAYS
9 = 13
sin 37° sin ∠PQR
  • Question 4

In  triangle PQR, which is not drawn to scale,

 

PQ = 9 cm, QR = 10 cm and ∠QRP = 37°

 

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

 

(Just write the number)

 

CORRECT ANSWER
101.0
EDDIE SAYS
9 = 10
sin 63° sin ∠QPR

∠PQR = 180° - 37° - ∠QPR
  • Question 5

In  triangle PQR, which is not drawn to scale,

 

PQ = 9 cm, QR = 10 cm and ∠QPR = 41°

 

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

 

(Just write the number)

 

CORRECT ANSWER
36.2
EDDIE SAYS
10 = 9
sin 41° sin ∠QRP
  • Question 6

In  triangle PQR, which is not drawn to scale,

 

PQ = 14 cm, QR = 10 cm and ∠QPR = 41°

 

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

 

(Just write the number)

 

CORRECT ANSWER
72.3
EDDIE SAYS
10 = 14
sin 41° sin ∠QRP

∠PQR = 180° - 41° - ∠QRP
  • Question 7

In  triangle PQR, which is not drawn to scale,

 

PQ = 14 cm, PR = 10 cm and ∠PRQ = 123°

 

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

 

(Just write the number)

 

CORRECT ANSWER
20.2
EDDIE SAYS
14 = 10
sin 123° sin ∠PQR

∠QPR = 180° - 123° - ∠PQR
  • Question 8

In  triangle PQR, which is not drawn to scale,

 

QR = 7.5 cm, PR = 10 cm and ∠PQR = 40°

 

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

 

(Just write the number)

 

CORRECT ANSWER
28.8
EDDIE SAYS
10 = 7.5
sin 40° sin ∠QPR
  • Question 9

In  triangle PQR, which is not drawn to scale,

 

PQ = 8 cm, PR = 11 cm and ∠PQR = 111°

 

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

 

(Just write the number)

 

CORRECT ANSWER
26.2
EDDIE SAYS
11 = 8
sin 111° sin ∠QRP

∠QPR = 180° - 111° - ∠QRP
  • Question 10

In  triangle PQR, which is not drawn to scale,

 

PQ = 6 cm, PR = 12 cm and ∠QRP = 30°

 

Use the sine rule to find ∠PQR.

 

(Just write the number)

 

CORRECT ANSWER
90
90.0
EDDIE SAYS
6 = 12
sin 30° sin ∠PQR

This is a right-angled triangle
---- OR ----

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