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Year 11
Higher

The sine rule

I can derive and use the formula for the sine rule.

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New
New
Year 11
Higher

The sine rule

I can derive and use the formula for the sine rule.

Download all resources
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Share activities with pupils
arrow-right
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Lesson details

Key learning points

  1. The sides of a triangle are proportional to the sines of their opposite angles
  2. A diagram can aid with understanding this
  3. The sine rule is useful when you know an angle and side pair and wish to find another pair
  4. In order to do this, you will need either a side length or an angle

Keywords

  • Sine rule - The sine rule is a formula used for calculating either an unknown side length or the size of an unknown angle.

Common misconception

Pupils may be uncertain about which angle pairs with which side.

The angle is paired with the side length that is opposite.

You may wish to display the sine rule along with an appropriate diagram on a separate board so that pupils can reference this throughout the lesson.
Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

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Starter quiz

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6 Questions

Q1.
Which of the following are equivalent to $$\sin(\theta°)=\frac{\text{opp}}{\text{hyp}}$$?
Correct answer: $$\text{hyp}\times\sin(\theta°)=\text{opp}$$
Correct answer: $$\theta°=\arcsin\left(\frac{\text{opp}}{\text{hyp}}\right)$$
Correct answer: $$\text{hyp}=\frac{\text{opp}}{\sin(\theta°)}$$
$$\text{hyp}=\frac{\sin(\theta°)}{\text{opp}}$$
Q2.
Using $$\sin(A)=\frac{p}{c}$$, identify all the rearrangements.
An image in a quiz
Correct answer: $$p = \sin(A) \times c$$
Correct answer: $$c=\frac{p}{\sin(A)}$$
$$p=\frac{c}{\sin(A)}$$
$$c=\frac{\sin(A)}{p}$$
Q3.
Using $$\sin(C)=\frac{p}{a}$$, identify all the rearrangements.
An image in a quiz
$$p=\frac{a}{\sin(C)}$$
Correct answer: $$a=\frac{p}{\sin(C)}$$
Correct answer: $$p = \sin(C) \times a$$
$$a=\frac{\sin(C)}{p}$$
Q4.
An isosceles triangle has lengths 10 cm, 10 cm and 16 cm. The area of the isosceles triangle is cm$$^2$$.
Correct Answer: 48
Q5.
An equilateral triangle has lengths of 10 cm. The area of the triangle, to 1 decimal place, is cm$$^2$$.
Correct Answer: 43.3
Q6.
A regular hexagon has lengths of 6 cm. The area of the hexagon, to the nearest integer, is cm$$^2$$.
Correct Answer: 94

Exit quiz

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6 Questions

Q1.
When do we use the sine sule?
Correct answer: For calculating either an unknown side length or the size of an unknown angle
For calculating the area of a triangle
For calculating the area of a circle
For ensuring we write signs properly
Q2.
Identify the correct ways to express $$p$$ using the diagram.
An image in a quiz
Correct answer: $$p = \sin(A) \times c$$
Correct answer: $$p = \sin(C) \times a$$
$$p=\frac{a}{\sin(C)}$$
$$p=\frac{c}{\sin(A)}$$
Q3.
Which of the following are correct versions of the sine rule?
Correct answer: $$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$$
Correct answer: $$\frac{a}{\sin(A)}=\frac{b}{\sin(B)}$$
Correct answer: $$\frac{a}{\sin(A)}=\frac{c}{\sin(C)}$$
Correct answer: $$\frac{b}{\sin(b)}=\frac{c}{\sin(C)}$$
$$\frac{a}{\sin(b)}=\frac{b}{\sin(C)}$$
Q4.
Length $$a$$ is opposite angle A. Work out the size of length $$a$$ giving your answer to 2 decimal places.
An image in a quiz
Correct Answer: 9.74 cm, 9.74
Q5.
Length $$b$$ is opposite angle B. Work out the size of length $$b$$ giving your answer to 2 decimal places.
An image in a quiz
Correct Answer: 13.1 cm, 13.1
Q6.
Work out the size of angle $$\alpha$$ giving your answer to 1 decimal place.
An image in a quiz
Correct Answer: 57.0, 57.0 degrees, 57.0°