Right-angled Trigonometry

In this page, we will learn about right-angled trigonometry. For many, this will not be a new topic, nor will it be a difficult one. However, it deserves some time, as it forms the important step to understanding most of the other work in the geometry and trigonometry topic. The challenging part is when we look at trigonometry in 3 dimensions, as it can be difficult to visualise the triangles needed. The emphasis of this page is looking at these 3-D problems.


Key Concepts

On this page, you should learn about

  • Using sine, cosine and tangent ratios to find the sides and angles of right-angled triangles.
  • Angles of elevation and depression..
  • The size of an angle between between a line and a plane.

Summary

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Test Yourself

Here is a quiz that practises using the three trigonometric ratios



START QUIZ!
Here is a quiz that gets you to think about trigonometry in 3 dimensions

START QUIZ!

Exam-style Questions

Question 1

A, B and C are points on horizontal ground.

C is due West of B. A is due South of B. AB = 60m

A flagpole stands vertically at B.

From A, the angle of elevation of the top of the flagpole is 11°.

From B, the angle of elevation of the top of the flagpole is 15°.

Calculate the distance AC giving your answer to 3 significant figures.

Hint

Full Solution

Question 2

The diagram shows a cuboid ABCDEFGH. AB = 8 cm, AE = 6 cm and BC = 15 cm.

a) Find the length of AC.

Give your answer correct to 3 significant figures

b) Find the size of the angle between the line EC and the plane ABCD.

Give your answer correct to 1 decimal place.

Hint

Full Solution

 

Question 3

ABCDEF is a prism in which the triangle BCF is the cross section.

BC = 12cm, EF = 16cm, angle CBF = 30° and angle FCB = 90°

The angle AF makes with the plane ABCD is \(\large \theta\).

Show that \(\large \tan\theta=\frac{\sqrt{3}}{5}\)

Hint

Full Solution

 

Question 4

ABCDEFG is a triangular prism.

AB = 12cm, AE = 8cm, EF = 18cm.

Angle BAE = 90°

G is the midpoint of BC.

Calculate the angle between EG and the plane ABCD.

Give your answer correct to 1 decimal place.

Hint

Full Solution

 

Question 5

ABCDEF is a prism.

AB = AE = BE = 6cm. BC = 10cm

Calculate

a) the length EC

b) the angle AEC

c) the angle between EC and the plane ABCD

Give lengths to 3 significant figures and angles to 1 decimal place.

Hint

Full Solution

MY PROGRESS

How much of Right-angled Trigonometry have you understood?