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.

The unit circle

Maths

Checking and securing understanding of similar triangles

Checking and securing understanding of Pythagoras' theorem

Scaling the right-angled triangle from the unit circle

Sine and cosine ratios

Tangent ratio

Using the sine ratio

Using the cosine ratio

Using the tangent ratio

Choosing the right trigonometric ratio

Choosing an appropriate method for finding lengths of a triangle

Problem solving with trigonometry

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

In the prior unit, pupils continued to build on their understanding of congruence and similarity with right-angled triangles. In this unit, similarity is used to extend a right-angled triangle in the unit circle to any other right-angled triangle.

In this unit pupils learn how to recognise the relationship between the unit circle and the trigonometric ratios of sine, cosine and tangent. Pupils then apply this knowledge to solve trigonometric problems.

Trigonometry

What is the value of θ for sin(θ) = cos(θ)?

What is the value of θ for sin(θ) = tan(θ)?

In a right triangle, if the hypotenuse is 37.5 cm and a second side is 30 cm, what is the length of the third side? (Use a calculator to help you.)

It is not possible to know whether the triangles are similar or not.

Which of the following statements is true for these triangles?

An isosceles triangle has an internal angle of 50ᵒ, what might the other two angles be?

In a right triangle, if the shortest sides are 36 cm and 48 cm, what is the length of the hypotenuse? (Use a calculator to help you.)

Would a triangle ABC with sides AB = 10cm, BC = 7.5cm, AC = 12.5cm be similar to the one shown in the diagram?

Yes, because their three angles correspond.

No, because you always need a side and two angles.

For this pair of triangles, can you determine whether they are similar without using side lengths?

The unit circle

Lesson outcome