Sine and cosine ratios
Lesson details
Learning outcome
I can derive the sine and cosine ratios from the sides of a right-angled triangle.
Key learning points
- When the hypotenuse has a length of one, the opposite side has length sin(θ)
- When the hypotenuse has a length of one, the adjacent side has length cos(θ)
- 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.
Teacher tip
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.
Licence
Lesson video
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Prior knowledge starter quiz
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.)
Q2.For this pair of triangles, can you determine whether they are similar without using side lengths?
Q3.What is cos(60ᵒ)?
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.)
Q5.Which of the following statements is true for these triangles?
Q6.Would a triangle ABC with sides AB = 32cm, BC = 24cm, AC = 40cm be similar to the one shown in the diagram?
Assessment exit quiz
6 Questions
Q1.Which side would we label as 'adjacent' to the non-right angle shown?
Q2.In a right triangle, if you are provided with θ and the adjacent side, which function would you use to find the hypotenuse?
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?
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?
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?
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?
To help you plan your 9 maths lesson on: Sine and cosine ratios, download all teaching resources for free and adapt to suit your pupils' needs...
To help you plan your 9 maths lesson on: Sine and cosine ratios, download all teaching resources for free and adapt to suit your pupils' needs.
The starter quiz will activate and check your pupils' prior knowledge, with versions available both with and without answers in PDF format.
We use learning cycles to break down learning into key concepts or ideas linked to the learning outcome. Each learning cycle features explanations with checks for understanding and practice tasks with feedback. All of this is found in our slide decks, ready for you to download and edit. The practice tasks are also available as printable worksheets and some lessons have additional materials with extra material you might need for teaching the lesson.
The assessment exit quiz will test your pupils' understanding of the key learning points.
Our video is a tool for planning, showing how other teachers might teach the lesson, offering helpful tips, modelled explanations and inspiration for your own delivery in the classroom. Plus, you can set it as homework or revision for pupils and keep their learning on track by sharing an online pupil version of this lesson.
Explore more key stage 3 maths lessons from the Trigonometry unit, dive into the full secondary maths curriculum, or learn more about lesson planning.