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Year 10
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Checking and securing understanding of the unit circle

I can see how the trigonometric functions are derived from measurements within a unit circle and how this can be utilised.

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Year 10
Higher

Checking and securing understanding of the unit circle

I can see how the trigonometric functions are derived from measurements within a unit circle and how this can be utilised.

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Lesson details

Key learning points

  1. Trigonometric functions are derived from measurements within a unit circle
  2. The right-angled triangle within the unit circle has a hypotenuse of length one unit
  3. The triangle can be scaled to any other right-angled triangle
  4. Similar triangles have the same interior angles
  5. Similar triangles have the same trigonometric ratios

Keywords

  • Trigonometric functions - Trigonometric functions are commonly defined as ratios of two sides of a right-angled triangle for a given angle.

  • Sine function - The sine of an angle (sin(θ°)) is the y-coordinate of point P on the triangle formed inside the unit circle.

  • Cosine function - The cosine of an angle (cos(θ°)) is the x-coordinate of point P on the triangle formed inside the unit circle.

  • Tangent function - The tangent of an angle (tan(θ°)) is the y-coordinate of point Q on the triangle which extends from the unit circle.

Common misconception

When reading the values of the trigonometric functions during the explanation slides and the tasks, pupils may think that all the values taken from the graphs are fully accurate.

Explain that many of the values from the trigonometric functions have digits beyond the second decimal place. Their calculator has these values stored to far more decimal places.


To help you plan your year 10 maths lesson on: Checking and securing understanding of the unit circle, download all teaching resources for free and adapt to suit your pupils' needs...

The slide deck contains several links to GeoGebra applets with the unit circle. Use these during explanations to show how properties within the unit circle change when the angle varies. Alternatively, if pupils have access to the internet, they could explore these for themselves.
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Licence

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).

Lesson video

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Starter quiz

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6 Questions

Q1.
A tangent to a curve at a given point is a line that intersects the curve at that point. Both the tangent and the curve have the same at the given point.
Correct Answer: gradient
Q2.
What are the coordinates of P?
An image in a quiz
(0.6, 0.8)
(0.8, -0.6)
(-0.6, 0.8)
Correct answer: (0.8, 0.6)
(-0.8, -0.6)
Q3.
What are the coordinates of Q?
An image in a quiz
(-0.4, -0.6)
Correct answer: (-0.6, -0.4)
(-0.6, 0.4)
(0.4, 0.6)
(0.4, -0.6)
Q4.
A right-angled triangle has a hypotenuse of 1 cm and a short side of 0.6 cm, what is the length of the third side of the triangle?
Correct Answer: 0.8 cm, 0.8
Q5.
A right-angled triangle has a hypotenuse of 1 cm and a short side of 0.54 cm, what is the length of the third side of the triangle, to 2 decimal places?
Correct Answer: 0.84 cm
Q6.
A right-angled triangle has a hypotenuse of 2.6 cm and a short side of 1 cm, what is the length of the third side of the triangle?
Correct Answer: 2.4 cm

Exit quiz

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6 Questions

Q1.
What are the features of the unit circle?
centred anywhere on the coordinate grid
Correct answer: centred at the origin
Correct answer: has a diameter of 2 units
has a radius of 1 centimetre
Q2.
The sine of an angle is the __________ of the point where the radius of the unit circle has been rotated through that angle.
$$x$$-coordinate
Correct answer: $$y$$-coordinate
Q3.
The __________ of an angle is the $$x$$-coordinate of the point where the radius of the unit circle has been rotated through that angle.
sine
Correct answer: cosine
tangent
Q4.
The tangent of an angle is the __________ of the point where the line (the triangle’s hypotenuse) intersects the tangent line.
$$x$$-coordinate
Correct answer: $$y$$-coordinate
Q5.
Match up the correct trigonometric values.
An image in a quiz
Correct Answer:$$\sin(50^\circ)$$,0.77
tick

0.77

Correct Answer:$$\cos(50^\circ)$$,0.64
tick

0.64

Correct Answer:$$\tan(50^\circ)$$,1.19
tick

1.19

Q6.
Using this diagram, what can be deduced?
An image in a quiz
$$\sin(\theta^\circ)=\cos(\theta^\circ)$$
$$\sin(\theta^\circ)+\cos(\theta^\circ)=1$$
Correct answer: $$(\sin(\theta^\circ))^2+(\cos(\theta^\circ))^2=1$$
Correct answer: $$-1\leq\cos(\theta^\circ)\leq1$$
$$\sin(\theta^\circ)>1$$