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

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.

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 $$\sin(\theta)\div\cos(\theta)$$.

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