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Year 9

Sine and cosine ratios

I can derive the sine and cosine ratios from the sides of a right-angled triangle.

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Year 9

Sine and cosine ratios

I can derive the sine and cosine ratios from the sides of a right-angled triangle.

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Lesson details

Key learning points

  1. When the hypotenuse has a length of one, the opposite side has length sin⁡(θ)
  2. When the hypotenuse has a length of one, the adjacent side has length cos⁡(θ)
  3. Any right-angled triangle is a scaled version of this triangle.

Keywords

  • Hypotenuse - The hypotenuse is the side of a right-angled triangle which is opposite the right angle.

  • Adjacent - The adjacent side of a right-angled triangle is the side which is next to both the right angle and the marked angle.

  • Opposite - The opposite side of a right-angled triangle is the side which is opposite the marked angle.

  • Trigonometric ratios - The trigonometric ratios are ratios between each pair of lengths in a right-angled triangle.

Common misconception

sin(60°) is double sin(30°).

The values of the sine or cosine of an angle do not scale linearly. We can see from the unit circle that an angle of 30° meets the circle at a height of 0.5 units, whilst an angle of 60° meets the circle at a height of approximately 0.87 units.


To help you plan your year 9 maths lesson on: Sine and cosine ratios, download all teaching resources for free and adapt to suit your pupils' needs...

Calculators can be used across all learning cycles to perform calculations using values found in the table of trig values or measured from the unit circle, however there is a bespoke learning cycle dedicated to introducing trigonometric values using a calculator.
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This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Starter quiz

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6 Questions

Q1.
In a right triangle, if the hypotenuse is 25 cm and a second side is 15 cm, what is the length of the third side? (Use a calculator to help you.)
Correct answer: 20 cm
22 cm
19 cm
Q2.
For this pair of triangles, can you determine whether they are similar without using side lengths?
An image in a quiz
Correct answer: Yes, because their three angles correspond.
No because you only know two angles.
No, because you always need a side and two angles.
Q3.
What is cos(60ᵒ)?
Correct answer: 0.5
0.1
0
1
Q4.
In a right triangle, if the shortest sides are 40 cm and 96 cm, what is the length of the hypotenuse? (Use a calculator to help you.)
100 cm
Correct answer: 104 cm
108 cm
Q5.
Which of the following statements is true for these triangles?
An image in a quiz
Correct answer: The triangles are similar.
The triangles are not similar.
It is not possible to know whether the triangles are similar or not.
Q6.
Would a triangle ABC with sides AB = 32cm, BC = 24cm, AC = 40cm be similar to the one shown in the diagram?
An image in a quiz
Correct answer: Yes
No

Exit quiz

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6 Questions

Q1.
Which side would we label as 'adjacent' to the non-right angle shown?
An image in a quiz
A
Correct answer: B
C
Q2.
In a right triangle, if you are provided with θ and the adjacent side, which function would you use to find the hypotenuse?
Sine
Correct answer: Cosine
Tangent
Q3.
In a right triangle, if θ = 35ᵒ and the opposite side is 4 cm, what is the length of the hypotenuse to the nearest whole number?
Correct Answer: 7 cm, 7
Q4.
In a right triangle, if you are provided with θ and the opposite side, which function would you use to find the adjacent side?
Sine
Cosine
Correct answer: Tangent
Q5.
In a right triangle, if θ = 40ᵒ and the adjacent side is 9 cm, what is the length of the hypotenuse to the nearest whole number?
Correct Answer: 12 cm, 12
Q6.
In a right triangle, if θ = 50ᵒ and the hypotenuse is 12 cm, what is the length of the adjacent side to the nearest whole number?
Correct Answer: 8 cm, 8

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