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

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

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

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

Calculate trigonometric ratios for 0°, 45° and 90°

Maths

Checking and further securing understanding of Pythagoras' theorem

Using Pythagoras' theorem to justify a right-angled triangle

Calculating the length of a line segment

Checking and securing understanding of the unit circle

Checking and securing understanding of sine ratio problems

Checking and securing understanding of cosine problems

Checking and securing understanding of tangent ratio problems

Calculate trigonometric ratios for 30° and 60°

Applying trigonometric ratios in context

Problem solving with right-angled trigonometry

In this unit, pupils have explored the trigonometric ratios for right-angled triangles in a variety of situations. In the future unit, pupils explore what can be done when we have a triangle that is not right-angled.

In the prior unit, pupils were introduced to the trigonometric ratios and developed the skills to ascertain which ratio would be appropriate for a given problem. In this unit, pupils are introduced to a variety of situations where this knowledge is required, such as 3-D problems and elevation/depression.

In this unit, pupils are introduced to a variety of situations where trigonometric ratios are required, such as 3-D problems and problems of elevation or depression.

Right-angled trigonometry

... two $$45^\circ$$ angles and a right-angle.

... a hypotenuse $$\sqrt{2}$$ times longer than the other two edges.

... two right-angles and one other angle.

What is the exact value of $$\cos(45^\circ)$$?

What is the exact value of $$\tan(45^\circ)$$?

undefined as the line does not intersect the tangent.

This diagram shows that $$\sin(90^\circ)=1$$ and $$\cos(90^\circ)=0$$, but what is $$\tan(90^\circ)$$?

What is the exact value of $$\tan(45^\circ)+\sin(90^\circ)$$?

Which trigonometric ratio is between the opposite and adjacent?

Which of these have a value of $$\frac{\sqrt{3}}{2}$$?

$$\tan(60^\circ)=\sqrt{3}$$, so what is $$(\tan(60^\circ))^2$$?

Sam 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?

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

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

$$=\frac{\text{opp}}{\text{hyp}}\div\frac{\text{adj}}{\text{hyp}}$$

$$=\frac{\text{opp}}{\text{hyp}}\times\frac{\text{hyp}}{\text{adj}}$$

Order the stages of working for $$\sin(\theta)\div\cos(\theta)$$.

Starter Quiz

Which trigonometric ratio is between the opposite and adjacent?