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Calculate trigonometric ratios for 0°, 45° and 90°
I can calculate trigonometric ratios for 0°, 45° and 90°.
Lesson 9 of 13
New
New
- Year 10
Calculate trigonometric ratios for 0°, 45° and 90°
I can calculate trigonometric ratios for 0°, 45° and 90°.
Lesson details
Key learning points
- The trigonometric ratios for 45° can be calculated using a square
- The square should have lengths of 1 unit
- By splitting the square into two right-angled triangles, you can calculate the ratio
- The ratios for 0° and 90° can be reasoned
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
Trigonometry always involves rounding.
Try evaluating sin(0) on your calculator. What answer do you get?
The first learning cycle is heavily scaffolded to support pupils who may find this difficult. You may wish to remove some of this so your pupils can do more deduction.
Teacher tip
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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Prior knowledge starter quiz
6 Questions
Q1.Which trigonometric ratio is between the opposite and adjacent?
Which trigonometric ratio is between the opposite and adjacent?
sine
cosine
Correct answer: tangent
tangent
Q2.Which of these have a value of $$\frac{\sqrt{3}}{2}$$?
Which of these have a value of $$\frac{\sqrt{3}}{2}$$?
$$\sin(30^\circ)$$
Correct answer: $$\cos(30^\circ)$$
$$\cos(30^\circ)$$
$$\tan(30^\circ)$$
Correct answer: $$\sin(60^\circ)$$
$$\sin(60^\circ)$$
$$\cos(60^\circ)$$
Q3.$$\tan(60^\circ)=\sqrt{3}$$, so what is $$(\tan(60^\circ))^2$$?
$$\tan(60^\circ)=\sqrt{3}$$, so what is $$(\tan(60^\circ))^2$$?
Correct Answer: 3, three
3, three
Q4.Sam has used their calculator to answer a trigonometry question. What is the answer a value of?
Sam has used their calculator to answer a trigonometry question. What is the answer a value of?
opposite
adjacent
Correct answer: hypotenuse
hypotenuse
angle
Q5.Aisha has used their calculator to answer a trigonometry question. What is the answer a value of?
Aisha has used their calculator to answer a trigonometry question. What is the answer a value of?
Correct answer: opposite
opposite
adjacent
hypotenuse
angle
Q6.Order the stages of working for $$\sin(\theta)\div\cos(\theta)$$.
Order the stages of working for $$\sin(\theta)\div\cos(\theta)$$.
1 - $$\sin(\theta)=\frac{\text{opp}}{\text{hyp}}$$
1
- $$\sin(\theta)=\frac{\text{opp}}{\text{hyp}}$$
2 - and $$\cos(\theta)=\frac{\text{adj}}{\text{hyp}}$$
2
- and $$\cos(\theta)=\frac{\text{adj}}{\text{hyp}}$$
3 - Hence, $$\sin(\theta)\div\cos(\theta)$$
3
- Hence, $$\sin(\theta)\div\cos(\theta)$$
4 - $$=\frac{\text{opp}}{\text{hyp}}\div\frac{\text{adj}}{\text{hyp}}$$
4
- $$=\frac{\text{opp}}{\text{hyp}}\div\frac{\text{adj}}{\text{hyp}}$$
5 - $$=\frac{\text{opp}}{\text{hyp}}\times\frac{\text{hyp}}{\text{adj}}$$
5
- $$=\frac{\text{opp}}{\text{hyp}}\times\frac{\text{hyp}}{\text{adj}}$$
6 - $$=\frac{\text{opp}}{\text{adj}}$$
6
- $$=\frac{\text{opp}}{\text{adj}}$$
7 - $$=\tan(\theta)$$.
7
- $$=\tan(\theta)$$.
Assessment exit quiz
6 Questions
Q1.A right-angled isosceles triangle has...
A right-angled isosceles triangle has...
Correct answer: ... two $$45^\circ$$ angles and a right-angle.
... two $$45^\circ$$ angles and a right-angle.
... three different edge lengths.
Correct answer: ... a hypotenuse $$\sqrt{2}$$ times longer than the other two edges.
... a hypotenuse $$\sqrt{2}$$ times longer than the other two edges.
... two right-angles and one other angle.
Q2.What is the exact value of $$\cos(45^\circ)$$?
What is the exact value of $$\cos(45^\circ)$$?
Correct answer: $$\frac{1}{\sqrt{2}}$$
$$\frac{1}{\sqrt{2}}$$
$$1$$
$$\sqrt{2}$$
Correct answer: $$\frac{\sqrt{2}}{2}$$
$$\frac{\sqrt{2}}{2}$$
Q3.What is the exact value of $$\tan(45^\circ)$$?
What is the exact value of $$\tan(45^\circ)$$?
$$\frac{1}{\sqrt{2}}$$
Correct answer: $$1$$
$$1$$
$$\sqrt{2}$$
$$\frac{\sqrt{2}}{2}$$
Q4.What does this diagram show?
What does this diagram show?
$$\sin(0^\circ)=1$$
Correct answer: $$\sin(0^\circ)=0$$
$$\sin(0^\circ)=0$$
$$\cos(0^\circ)=0$$
Correct answer: $$\cos(0^\circ)=1$$
$$\cos(0^\circ)=1$$
Correct answer: $$\tan(0^\circ)=0$$
$$\tan(0^\circ)=0$$
Q5.This diagram shows that $$\sin(90^\circ)=1$$ and $$\cos(90^\circ)=0$$, but what is $$\tan(90^\circ)$$?
This diagram shows that $$\sin(90^\circ)=1$$ and $$\cos(90^\circ)=0$$, but what is $$\tan(90^\circ)$$?
0 as there is not an answer.
Correct answer: undefined as the line does not intersect the tangent.
undefined as the line does not intersect the tangent.
1 as the tangent is $$x=1$$
Q6.What is the exact value of $$\tan(45^\circ)+\sin(90^\circ)$$?
What is the exact value of $$\tan(45^\circ)+\sin(90^\circ)$$?
Correct Answer: 2, two
2, two
