Mathematics Stage 5 – Trigonometry and The Unit Circle
Outcomes Assessed
MA5.3-2WM - generalises mathematical ideas and techniques to analyse and solve
problems efficiently
MA5.3-15MG - applies Pythagoras’ theorem, trigonometric relationships, the sine rule, the
cosine rule and the area rule to solve problems, including problems involving three
dimensions
Task 1 - The Unit Circle
(a) You are to construct a unit circle using the geometrical instruments. (4 marks)
Instructions:
 On an A4 graph paper, draw a number plane and construct a circle with a radius
of 10 cm.
 Label axes, origin, scales and major angles on the unit circle.
 Number the quadrants.
(b) Use the unit circle you have constructed to find the value of the following, correct to
2 decimal places. Leave all the traces (markings) of your work on the number plane
as an evidence of your working. (Do not use a calculator) (10 marks)
(i)
(ii)
(iii)
(iv)
(v)
sin 50⁰
cos 150⁰
tan 225⁰
cos 340⁰
sin 270⁰
(c) Use the unit circle to solve each equation for 0⁰ ≤ θ ≤ 360⁰. Find all possible angles.
Give your answer correct to the nearest degree. Show your working on the unit
circle. (6 marks)
(i)
(ii)
Sin θ = 0.8
Cos θ = -0.4
(d) Copy and complete the table shown below using the unit circle from Q1.
Then draw a neat graph of:
y = sin x for 0ᴼ ≤ x ≤ 360o. (3 marks)
y = tan x for 0ᴼ ≤ x ≤ 360o. (3 marks)
(i)
(ii)
x
0ᴼ
30ᴼ
45ᴼ
60ᴼ
90ᴼ
120ᴼ
240 ͦ
270 ͦ
300 ͦ
315 ͦ
330 ͦ
360 ͦ
135ᴼ
150 ͦ
180 ͦ
210 ͦ
225 ͦ
sin x
tan x
x
sin x
tan x
Task 2 - Trigonometric problem
Whilst walking due north Felicity turns at point A to avoid a lake. She then walks 250m to
point B on a bearing of 052°, she then changes direction and follows a new bearing of 120°
to point C. C is due east of A.
(i)
(ii)
(iii)
Draw a diagram using your knowledge of bearings. (1 marks)
By using basic geometric theorems, calculate all the angles in ∆ABC. (3 marks)
Find the distance AC using appropriate trigonometric formulas,
correct to the nearest metre. (3 marks)
Task 3 - Trigonometric Problem
Sara is standing on the shore at point A and observes a boat 300 metres away on a bearing
of 043°. While Amelia standing on the same shore at point B observes the same boat 580m
away and on a bearing of 300°.
(i)
(ii)
(iii)
Draw a diagram to illustrate this. (1 marks)
Find the distance between Sara and Amelia, correct to 1 decimal place. (3 marks)
Find the bearing of point B, where Amelia is, from point A, where Sara is.
Answer to the nearest degree. (3 marks)