New
New
Year 11
Higher

Checking and securing understanding of trigonometric ratios

I can appreciate the range of values of the trigonometric functions.

New
New
Year 11
Higher

Checking and securing understanding of trigonometric ratios

I can appreciate the range of values of the trigonometric functions.

Lesson details

Key learning points

  1. The unit circle is a circle with a radius of one
  2. The unit circle is centered on the origin
  3. The sine of an angle is the y-coordinate of the point where the radius has been rotated through that angle
  4. The cosine of an angle is the x-coordinate of the point where the radius has been rotated through that angle
  5. 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.

You may wish to extend pupils during the tasks by having them calculate all three sides and all the angles in the triangles in different ways (such as using Pythagoras' theorem or the other trigonometric formulas). This allows pupils to both check their work but also to appreciate the connectivity.
Teacher tip

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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6 Questions

Q1.
Trigonometric functions are commonly defined as ratios of two sides of a right-angled triangle for a given angle. Select the trigonometric functions.
Correct answer: Sine
Correct answer: Cosine
Correct answer: Tangent
Pythagoras' theorem
Reciprocal
Q2.
What functions do we use to find a missing angle when provided with two sides in a right-angled triangle?
Correct answer: arccos
Correct answer: arctan
Correct answer: arcsin
$$(\cos(\theta°))^{-1}$$
$$(\sin(\theta°))^{-1}$$
Q3.
Which of the following calculations will calculate the answer to the length $$e$$?
An image in a quiz
Correct answer: $$e =\frac{\text{9}}{\cos(52°)}$$
Correct answer: $$e =\frac{\text{9}}{\sin(38°)}$$
$$e =\frac{\sin(38°)}{\text{9}}$$
$$e =\frac{\cos(52°)}{\text{9}}$$
Q4.
Which of the following are equivalent to $$\cos(\theta°)=\frac{\text{adj}}{\text{hyp}}$$?
Correct answer: $$\theta°=\arccos\left(\frac{\text{adj}}{\text{hyp}}\right)$$
Correct answer: $$\text{hyp}\times\cos(\theta°)=\text{adj}$$
Correct answer: $$\text{hyp}=\frac{\text{adj}}{\cos(\theta°)}$$
$$\cos(\theta°)=\frac{\text{hyp}}{\text{adj}}$$
Q5.
Which of the following are equivalent to $$\tan(\theta°)=\frac{\text{opp}}{\text{adj}}$$?
Correct answer: $$\text{adj}\times\tan(\theta°)=\text{opp}$$
$$\tan(\theta°)=\frac{\text{adj}}{\text{opp}}$$
Correct answer: $$\text{adj}=\frac{\text{opp}}{\tan(\theta°)}$$
Correct answer: $$\theta°=\arctan\left(\frac{\text{opp}}{\text{adj}}\right)$$
Q6.
Which of the following are equivalent to $$\sin(\theta°)=\frac{\text{opp}}{\text{hyp}}$$?
$$\text{hyp}=\frac{\sin(\theta°)}{\text{opp}}$$
Correct answer: $$\theta°=\arcsin\left(\frac{\text{opp}}{\text{hyp}}\right)$$
Correct answer: $$\text{hyp}=\frac{\text{opp}}{\sin(\theta°)}$$
Correct answer: $$\text{hyp}\times\sin(\theta°)=\text{opp}$$

6 Questions

Q1.
What trigonometric ratio would you use to work out the length $$k$$?
An image in a quiz
Sine
Cosine
Correct answer: Tangent
Pythagoras
Q2.
What trigonometric ratio would you use to work out the length $$b$$?
An image in a quiz
Correct answer: Sine
Cosine
Tangent
Pythagoras
Either Sine or Cosine
Q3.
What trigonometric ratio would you use to work out the length $$e$$?
An image in a quiz
Correct answer: Sine or cosine
Sine only
Tangent only
Cosine or tangent
Pythagoras
Q4.
Work out the length $$b$$ to 1 decimal place.
An image in a quiz
Correct Answer: 5.6, 5.6 cm
Q5.
Work out the length $$e$$ to 1 decimal place.
An image in a quiz
Correct Answer: 33.5 , 33.5 cm
Q6.
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 $$c$$.
Correct Answer:50.2°,When $$a$$ = 6 cm and $$b$$ = 5 cm, angle $$x$$° is ...

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

Correct Answer:30°,When $$a$$ = 6 cm and $$c$$ = 12 cm, angle $$x$$° is ...

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

Correct Answer:60°,When $$b$$ = 4 cm and $$c$$ = 8 cm, angle $$x$$° is ...

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

Correct Answer:63.4°,When $$a$$ = 2 cm and $$b$$ = 1 cm, angle $$x$$° is ...

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