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The sine rule practice questions with answers

Higher · Grade 7 · Calculator

A second set of the sine rule questions for GCSE Maths. They are different from the ones on the topic page, so work through both. Try each question, then open the answer to check your working.

More questions

Question 1 Easy

In triangle ABC, side a is 8 cm, side b is 5 cm and angle A is 50 degrees. Angle B is acute. Find angle B to the nearest degree.

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Answer: 29

  1. sin B = 5 sin 50 / 8.
  2. The acute angle is 29 degrees.
  • Method (1): sin B = 5 sin 50 / 8.
  • Accuracy (1): Answer 29

Question 2 Easy

Use the sine rule. The side opposite 30 degrees is 5. Find the side opposite 60 degrees.

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Answer: 5√3

  1. sin 60 / sin 30 = √3.
  2. 5 × √3 = 5√3.
  • Method (1): sin 60 / sin 30 = √3.
  • Accuracy (1): Answer 5√3

Question 3 Medium

From port P, ship A is on a bearing of 40 degrees and ship B is on a bearing of 110 degrees. PA is 9 km and angle PAB is 40 degrees. Find PB to 3 significant figures.

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Answer: 6.16

  1. The angle at P is 110 − 40 = 70 degrees.
  2. PB = 9 sin 40 / sin 70.
  • Method (1): The angle at P is 110 − 40 = 70 degrees.
  • Accuracy (1): Answer 6.16

Question 4 Medium

Use the sine rule. The side opposite 60 degrees is 6√3. Find the side opposite 30 degrees.

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Answer: 6

  1. sin 30 / sin 60 = 1/√3.
  2. 6√3 / √3 = 6.
  • Method (1): sin 30 / sin 60 = 1/√3.
  • Accuracy (1): Answer 6

Question 5 Hard

In triangle ABC, side a is 10 cm, side b is 7 cm and angle A is 55 degrees. Angle B is acute. Find angle B to the nearest degree.

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Answer: 35

  1. sin B = 7 sin 55 / 10.
  2. Take the acute angle.
  • Method (1): sin B = 7 sin 55 / 10.
  • Accuracy (1): Answer 35

Question 6 Hard

In triangle ABC, angle A is 50 degrees and angle B is 60 degrees. Side AB is 14 cm. Find side BC to 3 significant figures.

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Answer: 11.4

  1. Angle C is 70 degrees.
  2. BC = 14 sin 50 / sin 70.
  • Method (1): Angle C is 70 degrees.
  • Accuracy (1): Answer 11.4

Exam-style question

A longer question with marks, in the style of a written paper. This is a Yumio practice question, not a past paper question.

Exam-style question 12 marks

Use the sine rule. The side opposite 30 degrees is 13. Find the side opposite 60 degrees.

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Answer: 13√3

  1. The 60 degree side is √3 times the 30 degree side.
  2. 13√3.
  • Method (1): The 60 degree side is √3 times the 30 degree side.
  • Accuracy (1): Answer 13√3

Quick checks

Short one-step questions. Pick an option before you open the answer.

Check 1

What does the sine rule pair together?

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A side and the angle opposite that side. A side and the angle opposite that side.

Check 2

Side opposite 30 degrees is 5. What side is opposite 45?

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Sin 45 / sin 30 = √2, so side is 5√2. Sin 45 / sin 30 = √2, so side is 5√2.

Check 3

Why is the scale from 30 to 45 equal to √2?

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(√2/2) / (1/2) = √2. (√2/2) / (1/2) = √2.

Check 4

Side opposite 45 degrees is 5√2. What side is opposite 30?

  1. 2
  2. 5
  3. 10
Show answer

5 Divide by √2: 5√2 / √2 = 5.

Check 5

Side opposite 30 is 5 and unknown acute side is 5√2. What is the angle?

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Sin of angle is √2/2, so angle is 45 degrees. Sin of angle is √2/2, so angle is 45 degrees.

Check 6

Angles 50 and 60 are in a triangle. What is the third?

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180 - 50 - 60 = 70 degrees. 180 - 50 - 60 = 70 degrees.

Check 7

From 180, take away 40 and 70. What angle is left?

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180 - 40 - 70 = 70 degrees. 180 - 40 - 70 = 70 degrees.

Check 8

From 180, take away 55 and 65. What angle is left?

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180 - 55 - 65 = 60 degrees. 180 - 55 - 65 = 60 degrees.

Check 9

Why mention 50, 60 and 70?

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The sine rule works on any general triangle. The sine rule works on any general triangle.

Check 10

A student pairs a side with adjacent angle. What should change?

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Pair each side with the angle facing it. Pair each side with the angle facing it.

Check 11

Side opposite 30 degrees is 10. What is opposite 45?

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10 × √2 = 10√2. 10 × √2 = 10√2.

Check 12

Side opposite 45 degrees is 8√2. What is opposite 30?

  1. 16
  2. 45
  3. 8
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8 8√2 / √2 = 8.

Check 13

Angles 50 and 80 leave what third angle?

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180 - 50 - 80 = 50 degrees. 180 - 50 - 80 = 50 degrees.

Check 14

The sine of an angle is √2/2. Which angle does this lesson ask for?

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The acute angle is 45 degrees. Its obtuse partner is 180 - 45 = 135 degrees. The acute angle, 45 degrees.

Check 15

What is 180 minus 50 minus 60?

  1. 50
  2. 60
  3. 70
Show answer

70 180 - 110 = 70.

Keep going with The sine rule

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