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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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New
New
Year 9

Sine and cosine ratios

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

Download all resources
Share activities with pupils

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.

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.

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.

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).

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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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