<p> Mathematics Stage 5 – Trigonometry and The Unit Circle</p><p>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</p><p>Task 1 - The Unit Circle</p><p>(a) You are to construct a unit circle using the geometrical instruments. (4 marks)</p><p>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.</p><p>(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)</p><p>(i) sin 50⁰ (ii) cos 150⁰ (iii) tan 225⁰ (iv) cos 340⁰ (v) sin 270⁰</p><p>(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)</p><p>(i) Sin θ = 0.8 (ii) Cos θ = -0.4 (d) Copy and complete the table shown below using the unit circle from Q1. Then draw a neat graph of:</p><p>(i) y = sin x for 0ᴼ ≤ x ≤ 360o. (3 marks) (ii) y = tan x for 0ᴼ ≤ x ≤ 360o. (3 marks)</p><p> x 0ᴼ 30ᴼ 45ᴼ 60ᴼ 90ᴼ 120ᴼ 135ᴼ 150 180 210 225 sin x tan x</p><p> x 240 270 300 315 330 360 sin x tan x</p><p>Task 2 - Trigonometric problem</p><p>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. </p><p>(i) Draw a diagram using your knowledge of bearings. (1 marks) (ii) By using basic geometric theorems, calculate all the angles in ∆ABC. (3 marks) (iii) Find the distance AC using appropriate trigonometric formulas, correct to the nearest metre. (3 marks)</p><p>Task 3 - Trigonometric Problem</p><p>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°. </p><p>(i) Draw a diagram to illustrate this. (1 marks) (ii) Find the distance between Sara and Amelia, correct to 1 decimal place. (3 marks) (iii) Find the bearing of point B, where Amelia is, from point A, where Sara is. Answer to the nearest degree. (3 marks)</p>