Checking and securing understanding of trigonometric ratios | Maths

Starter quiz

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

Sine ✓

Cosine ✓

Tangent ✓

Pythagoras' theorem

Reciprocal
What functions do we use to find a missing angle when provided with two sides in a right-angled triangle?

arccos ✓

arctan ✓

arcsin ✓

$(\cos(\theta°))^{-1}$

$(\sin(\theta°))^{-1}$
Which of the following calculations will calculate the answer to the length $e$ ?

$e =\frac{\text{9}}{\cos(52°)}$ ✓

$e =\frac{\text{9}}{\sin(38°)}$ ✓

$e =\frac{\sin(38°)}{\text{9}}$

$e =\frac{\cos(52°)}{\text{9}}$
Which of the following are equivalent to $\cos(\theta°)=\frac{\text{adj}}{\text{hyp}}$ ?

$\theta°=\arccos\left(\frac{\text{adj}}{\text{hyp}}\right)$ ✓

$\text{hyp}\times\cos(\theta°)=\text{adj}$ ✓

$\text{hyp}=\frac{\text{adj}}{\cos(\theta°)}$ ✓

$\cos(\theta°)=\frac{\text{hyp}}{\text{adj}}$
Which of the following are equivalent to $\tan(\theta°)=\frac{\text{opp}}{\text{adj}}$ ?

$\text{adj}\times\tan(\theta°)=\text{opp}$ ✓

$\tan(\theta°)=\frac{\text{adj}}{\text{opp}}$

$\text{adj}=\frac{\text{opp}}{\tan(\theta°)}$ ✓

$\theta°=\arctan\left(\frac{\text{opp}}{\text{adj}}\right)$ ✓
Which of the following are equivalent to $\sin(\theta°)=\frac{\text{opp}}{\text{hyp}}$ ?

$\text{hyp}=\frac{\sin(\theta°)}{\text{opp}}$

$\theta°=\arcsin\left(\frac{\text{opp}}{\text{hyp}}\right)$ ✓

$\text{hyp}=\frac{\text{opp}}{\sin(\theta°)}$ ✓

$\text{hyp}\times\sin(\theta°)=\text{opp}$ ✓

Exit quiz

What trigonometric ratio would you use to work out the length $k$ ?

Sine

Cosine

Tangent ✓

Pythagoras
What trigonometric ratio would you use to work out the length $b$ ?

Sine ✓

Cosine

Tangent

Pythagoras

Either Sine or Cosine
What trigonometric ratio would you use to work out the length $e$ ?

Sine or cosine ✓

Sine only

Tangent only

Cosine or tangent

Pythagoras
Work out the length $b$ to 1 decimal place.

'5.6' ✓
Work out the length $e$ to 1 decimal place.

'33.5 ' ✓
A right angled triangle is drawn and an angle $x$ is indicated. The opposite length to $x$ is labelled $a$ , the adjacent length to $x$ is $b$ and the hypotenuse is $c$ .

50.2°

⇔

When $a$ = 6 cm and $b$ = 5 cm, angle $x$ ° is ... ✓

30°

⇔

When $a$ = 6 cm and $c$ = 12 cm, angle $x$ ° is ... ✓

60°

⇔

When $b$ = 4 cm and $c$ = 8 cm, angle $x$ ° is ... ✓

63.4°

⇔

When $a$ = 2 cm and $b$ = 1 cm, angle $x$ ° is ... ✓

Worksheet

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Lesson Details

Key learning points

The unit circle is a circle with a radius of one
The unit circle is centered on the origin
The sine of an angle is the y-coordinate of the point where the radius has been rotated through that angle
The cosine of an angle is the x-coordinate of the point where the radius has been rotated through that angle
The tangent of an angle is the length of the side opposite the angle along the tangent at x = 1 to the unit circle

Common misconception

Pupils may use the incorrect trigonometric formula.

Encourage pupils to label their triangles with the name for each side. This helps to identify the opposite, adjacent and hypotenuse.

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.