Checking and securing understanding of the unit circle | Maths

Starter quiz

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.

'gradient' ✓
What are the coordinates of P?

(0.6, 0.8)

(0.8, -0.6)

(-0.6, 0.8)

(0.8, 0.6) ✓

(-0.8, -0.6)
What are the coordinates of Q?

(-0.4, -0.6)

(-0.6, -0.4) ✓

(-0.6, 0.4)

(0.4, 0.6)

(0.4, -0.6)
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?

'0.8 cm' ✓
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?

'0.84 cm' ✓
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?

'2.4 cm' ✓

Exit quiz

What are the features of the unit circle?

centred anywhere on the coordinate grid

centred at the origin ✓

has a diameter of 2 units ✓

has a radius of 1 centimetre
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

$y$ -coordinate ✓
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

cosine ✓

tangent
The tangent of an angle is the __________ of the point where the line (the triangle’s hypotenuse) intersects the tangent line.

$x$ -coordinate

$y$ -coordinate ✓
Match up the correct trigonometric values.

$\sin(50^\circ)$

⇔

0.77 ✓

$\cos(50^\circ)$

⇔

0.64 ✓

$\tan(50^\circ)$

⇔

1.19 ✓
Using this diagram, what can be deduced?

$\sin(\theta^\circ)=\cos(\theta^\circ)$

$\sin(\theta^\circ)+\cos(\theta^\circ)=1$

$(\sin(\theta^\circ))^2+(\cos(\theta^\circ))^2=1$ ✓

$-1\leq\cos(\theta^\circ)\leq1$ ✓

$\sin(\theta^\circ)>1$

Worksheet

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Lesson Details

Key learning points

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

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.

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.