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SCHOLAR Study Guide National 5 Mathematics Course Materials Topic 21: Trigonometric graphs and identities Authored by: Margaret Ferguson Reviewed by: Jillian Hornby Previously authored by: Eddie Mullan Heriot-Watt University Edinburgh EH14 4AS, United Kingdom. First published 2014 by Heriot-Watt University. This edition published in 2016 by Heriot-Watt University SCHOLAR. Copyright © 2016 SCHOLAR Forum. Members of the SCHOLAR Forum may reproduce this publication in whole or in part for educational purposes within their establishment providing that no profit accrues at any stage, Any other use of the materials is governed by the general copyright statement that follows. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, without written permission from the publisher. Heriot-Watt University accepts no responsibility or liability whatsoever with regard to the information contained in this study guide. Distributed by the SCHOLAR Forum. SCHOLAR Study Guide Course Materials Topic 21: National 5 Mathematics 1. National 5 Mathematics Course Code: C747 75 Acknowledgements Thanks are due to the members of Heriot-Watt University's SCHOLAR team who planned and created these materials, and to the many colleagues who reviewed the content. We would like to acknowledge the assistance of the education authorities, colleges, teachers and students who contributed to the SCHOLAR programme and who evaluated these materials. Grateful acknowledgement is made for permission to use the following material in the SCHOLAR programme: The Scottish Qualifications Authority for permission to use Past Papers assessments. The Scottish Government for financial support. The content of this Study Guide is aligned to the Scottish Qualifications Authority (SQA) curriculum. All brand names, product names, logos and related devices are used for identification purposes only and are trademarks, registered trademarks or service marks of their respective holders. 1 Topic 1 Trigonometric equations, graphs and identities Contents 21.1 Sketching trigonometric graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 3 21.2 Identifying trigonometric functions from graphs . . . . . . . . . . . . . . . . . . 21.3 Solving trigonometric equations . . . . . . . . . . . . . . . . . . . . . . . . . . 21 32 21.4 Exact trigonometric values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Using trigonometric identities . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 46 21.6 Learning Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.7 End of topic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 52 2 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Learning objectives By the end of this topic, you should be able to: • identify the features of Sine, Cosine and Tangent graphs; • identify the amplitude and period of a trig function; • identify and sketch trig graphs with: ◦ multiple angles; ◦ vertical translations; ◦ horizontal translations; • solve trigonometric equations; • determine exact trigonometric values; • use trig identities to simplify trigonometric expressions. © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES 1.1 3 Sketching trigonometric graphs Graphs of trigonometric functions (1) This activity allows you to investigate the graphs of various types of trigonometric functions. This will help you to be able to sketch the graphs of trigonometric functions from their equation. Using the equation y = a sin bx ◦ practice drawing graphs changing the values of a, b and the type of trigonometric function (sin, cos and tan). Here are some examples when the trigonometric function is sin. Notice how the values of a and b change the graph of the function. © H ERIOT-WATT U NIVERSITY Go online 4 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES .......................................... Key point The Amplitude of a graph The amplitude of a graph is half of the distance between the maximum and minimum values of the graph. For example: In y = a sin x and y = a cos x the amplitude is represented by a. © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES 5 Key point The Period of a graph The period of a graph is the distance along the x-axis over which the graph completes one full wave pattern. The period of y = sin x and y = cos x is 360 degrees. Key point y = tan x is different. . . The graph is not a smooth curve. The period of y = tan x is 180 ◦ . Graphs of trigonometric functions practice Q1: Sketch the graph of y = sin x. .......................................... Q2: Sketch the graph of y = cos x. .......................................... Q3: Sketch the graph of y = tan x. .......................................... Examples 1. Problem: Sketch the graph of y = sin 3x◦ for 0 ≤ x ≤ 360. Solution: Step 1: We want a ’sin’ wave which starts at the origin and rises to a maximum. © H ERIOT-WATT U NIVERSITY Go online 6 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Step 2: This maximum is 1 and the minimum is -1 since we want y = 1sin 3x ◦ . The amplitude = 1. Step 3: We need 3 waves in 360 degrees for y = sin 3x ◦ . Step 4: The period = 360 ÷ 3 = 120◦ so each full wave covers a distance of 120 ◦ . © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES .......................................... 2. Problem: Sketch the graph of y = 4 cos 2x◦ for 0 ≤ x ≤ 360. Solution: Step 1: We want a ’cos’ wave which starts at the maximum and drops through zero to the minimum. Step 2: This maximum is 4 and the minimum is -4 since we want y = 4 cos 2x ◦ . The amplitude = 4 Step 3: We need 2 waves in 360 degrees for y = 4 cos 2x ◦ . © H ERIOT-WATT U NIVERSITY 7 8 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Step 4: The period = 360 ÷ 2 = 180◦ so each full wave covers a distance of 180 ◦ . .......................................... 3. Problem: Sketch the graph of y = − tan 2x◦ for 0 ≤ x ≤ 360. Solution: Step 1: We need a ’tan’ graph which starts at the origin and has asymptotes at x = 90 ◦ and x = 270◦ . An asymptote is a line which the graph gets closer and closer to but never actually touches. The asymptotes will help us to make a better sketch. © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES 9 Step 2: The negative reflects the graph in the x-axis. Step 3: We require two copies of this tan graph in 360 ◦ for y = − tan 2x. The period of y = tan x is 180 ◦ . The period of y = − tan 2x is 180 ÷ 2 = 90 ◦ . We now have asymptotes at x = 45, 135, 225 and 315. .......................................... Key point The graph of y = − sin x is the graph of y = sin x reflected in the x-axis. Sketching trigonometric functions exercise 1 Q4: Sketch the graph of y = cos 4x◦ , for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a cos bx. .......................................... Q5: Sketch the graph of y = 3 sin x ◦ , for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a sin bx. .......................................... Q6: Sketch the graph of y = 5 tan 3x ◦ , for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a tan bx. .......................................... © H ERIOT-WATT U NIVERSITY Go online 10 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Q7: Sketch the graph of y = 4 cos 2x◦ , for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a cos bx. .......................................... Q8: Sketch the graph of y = 6 sin 4x ◦ , for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a sin bx. .......................................... Graphs of trigonometric functions (2) This activity allows you to investigate vertical translations of trigonometric functions. Go online Using the equation y = a sin bx ◦ + c practice drawing graphs changing the values of a, b, c and the type of trigonometric function (sin, cos and tan). Here are some examples when the trigonometric function is sin. Notice how the values of a, b and c change the graph of the function. © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES © H ERIOT-WATT U NIVERSITY 11 12 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Key point The graph of y = sin x + c is the graph of y = sin x moved up vertically by c units. The graph of y = sin x − c is the graph of y = sin x moved down vertically by c units. The same movement will be applied for y = cos x and y = tan x. .......................................... Examples 1. Problem: Sketch the graph of y = sin 2x◦ − 1, for 0 ≤ x ≤ 360◦ . Solution: Step 1: Start by sketching the graph of y = sin 2x◦ . The graph of y = sin 2x ◦ starts at the origin and rises to a maximum. The graph of y = sin 2x ◦ has a maximum of 1 and a minimum of -1. The amplitude = 1. There are 2 waves in y = sin 2x ◦ . The period = 360 ÷ 2 = 180◦ so each full wave covers a distance of 180 ◦ . Step 2: The graph of y = sin 2x ◦ − 1 is the graph of y = sin 2x◦ moved down vertically by 1 unit. © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES It is much easier to draw a faint version of the graph of y = sin 2x first then make a copy moved down vertically by 1 unit. The maximum is 0 and the minimum is -2. The amplitude = 1 as before. The period = 360 ÷ 2 = 180◦ as before. .......................................... 2. Problem: Sketch the graph of y = 2 cos 3x◦ + 4, for 0 ≤ x ≤ 360◦ Solution: Step 1: Start by sketching the graph of y = 2 cos 3x◦ . The graph of y = 2 cos 3x ◦ starts at 2 and decreases to a minimum. The graph of y = 2 cos 3x ◦ has a maximum of 2 and a minimum of -2. The amplitude = 2. There are 3 waves in y = 2 cos 3x ◦ . The period = 360 ÷ 3 = 120◦ so each full wave covers a distance of 120 ◦ . © H ERIOT-WATT U NIVERSITY 13 14 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Step 2: The graph of y = 2 cos 3x◦ + 4 is the graph of y = 2 cos 3x◦ moved up vertically by 4 units. The maximum is 6 and the minimum is 2. The amplitude = 2 as before. The period = 360 ÷ 3 = 120◦ as before. .......................................... 3. Problem: Sketch the graph of y = tan x ◦ + 3, for 0 ≤ x ≤ 360◦ Solution: Step 1: Start by sketching the graph of y = tan x ◦ . The graph of y = tan x ◦ starts at the origin and increases infinitely. The graph of y = tan x ◦ has no maximum or minimum. There is no amplitude. The tan graph is different from sin and cos. There are 2 waves in y = tan x ◦ . The period = 360 ÷ 2 = 180◦ so each full wave covers a distance of 180 ◦ . The graph has asymptotes where the it tends to positive and negative infinity. y = tan x◦ has asymptotes at 90◦ and 270◦ . © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES 15 Step 2: The graph of y = tan x ◦ + 3 is the graph of y = tan x◦ moved up vertically by 3 units. The period = 360 ÷ 2 = 180◦ as before. The asymptotes are at 90 ◦ and 270◦ as before. .......................................... Key point An asymptote is a vertical line of the form x = k at which the function is undefined. The graph gets closer and closer to the asymptote but never actually touches it. Sketching trigonometric functions exercise 2 Q9: Sketch the graph of y = cos 2x◦ + 2, for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a cos bx + c. .......................................... Q10: Sketch the graph of y = 5 sin 4x ◦ + 1, for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a sin bx + c. .......................................... Q11: Sketch the graph of y = tan 2x ◦ − 4, for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a tan bx + c. .......................................... Q12: Sketch the graph of y = 3 sin 2x ◦ − 4, for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a sin bx + c. .......................................... Q13: Sketch the graph of y = cos 4x ◦ − 2, for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a cos bx + c. .......................................... © H ERIOT-WATT U NIVERSITY Go online 16 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Graphs of trigonometric functions (3) This activity allows you to investigate horizontal translations of trigonometric functions. Go online Using the equation y = a sin (x + b ◦ ) + c practice drawing graphs changing the values of a, b, c and the type of trigonometric function (sin, cos and tan). Here are some examples when the trigonometric function is sin. Notice how the values of a, b and c change the graph of the function. © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES .......................................... Examples 1. Problem: Sketch the graph of y = sin (x◦ − 45), for 0 ≤ x ≤ 360◦ Solution: Step 1: Start by sketching the graph of y = sin x◦ . The graph of y = sin x ◦ starts at the origin and rises to a maximum. The graph of y = sin x ◦ has a maximum of 1 and a minimum of -1. The amplitude = 1. © H ERIOT-WATT U NIVERSITY 17 18 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES There is 1 wave in y = sin x◦ so the period = 360◦ . Step 2: The graph of y = sin (x ◦ − 45) is the graph of y = sin x◦ moved right by 45◦ . The maximum, minimum, amplitude and period are unchanged. It is much easier to draw a faint version of the graph of y = sin x ◦ first then make a copy 45◦ to the right. .......................................... 2. Problem: Sketch the graph of y = 3 cos (x◦ + 30), for 0 ≤ x ≤ 360◦ Solution: Step 1: Start by sketching the graph of y = 3 cos x◦ . The graph of y = 3 cos x ◦ starts at 3 and decreases to a minimum. The graph of y = 3 cos x ◦ has a maximum of 3 and a minimum of -3. The amplitude = 3. There is 1 wave in y = 3 cos x◦ so the period = 360◦ . © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Step 2: The graph of y = 3 cos (x◦ + 30) is the graph of y = 3 cos x◦ moved left by 30◦ . The maximum, minimum, amplitude and period are unchanged. It is much easier to draw a faint version of the graph of y = 3 cos x ◦ first then make a copy 30◦ to the left. .......................................... 3. Problem: Sketch the graph of y = 2 sin (x + 90◦ ) + 1, for 0 ≤ x ≤ 360◦ Solution: Step 1: Start by sketching the graph of y = 2 sin x◦ . The graph of y = 2 sin x ◦ starts at the origin and increases to a maximum. The graph of y = 2 sin x ◦ has a maximum of 2 and a minimum of -2. The amplitude = 2. There is 1 wave in y = 2 sin x◦ so the period = 360◦ . © H ERIOT-WATT U NIVERSITY 19 20 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Step 2: The graph of y = 2 sin x ◦ + 1 is the graph of y = 2 sin x◦ moved up vertically by 1 unit. The graph of y = 2 sin x ◦ + 1 has a maximum of 3 and a minimum of -1. The amplitude = 2 as before and the period is unchanged. Step 3: The graph of y = 2 sin (x + 90◦ ) + 1 is the graph of y = 2 sin x◦ + 1 moved left by 90◦ . Note again that it is much easier to make faint copies of y = 2 sin x ◦ and y = 2 sin x◦ + 1 then make a copy 90 ◦ to the left. Notice that the graph now looks like y = 2 cos x ◦ + 1. .......................................... © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES 21 Key point The graph of y = sin(x + c) is the graph of y = sin x moved c degrees to the left. The graph of y = sin(x − c) is the graph of y = sin x moved c degrees to the right. The same movement will be applied for y = cos x and y = tan x. Sketching trigonometric functions exercise 3 Q14: Sketch the graph of y = 6 sin (x ◦ − 15) − 2, for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a sin (x + b) + c. Go online .......................................... Q15: Sketch the graph of y = 2 cos (4x ◦ + 60) − 2, for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a cos (x + b) + c. .......................................... Q16: Sketch the graph of y = sin (3x ◦ + 60) + 4, for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a sin (x + b) + c. .......................................... Q17: Sketch the graph of y = tan (2x ◦ + 90) − 4, for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a tan (x + b) + c. .......................................... Q18: Sketch the graph of y = 3 cos (2x ◦ + 90) + 3, for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a cos (x + b) + c. .......................................... Q19: Sketch the graph of y = 3 tan (x ◦ + 45) − 1, for 0 ≤ x ≤ 360◦ . Remember the equation takes the form y = a tan (x + b) + c. .......................................... 1.2 Identifying trigonometric functions from graphs Before we identify the equations of trigonometric graphs use the activity below remind yourself what the graphs of some trigonometric functions look like. Graphs of trigonometric functions 1 Using the equation y = a sin bx practice drawing graphs changing the values of a and b and the type of trigonometric function (sin, cos and tan). Here are some examples, notice how the values of a and b change the graph of the function. © H ERIOT-WATT U NIVERSITY Go online 22 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES .......................................... Example Problem: Here is a sketch of the graph of y = a sin bx◦ , 0 ≤ x ≤ 360. What are the values of a and b? © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES 23 Solution: Step 1: The graph goes between a maximum of 3 and a minimum of -3 so the amplitude = 3, hence a = 3. Step 2: There are two complete waves in 360 ◦ so b = 2. (Alternatively since one wave = 180 ◦ , b = 360 ÷ 180 = 2). Hence the equation of the graph is y = 3 sin 2x ◦ . .......................................... Identifying trigonometric functions exercise 1 Q20: Identify the trigonometric function, y = a sin bx◦ , 0 ≤ x ≤ 360, shown in the graph. .......................................... © H ERIOT-WATT U NIVERSITY Go online 24 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Q21: Identify the trigonometric function, y = a cos bx◦ , 0 ≤ x ≤ 360, shown in the graph. .......................................... Q22: Identify the trigonometric function, y = a sin bx◦ , 0 ≤ x ≤ 360, shown in the graph. .......................................... Q23: Identify the trigonometric function, y = a cos bx◦ , 0 ≤ x ≤ 360, shown in the graph. .......................................... Before we identify the equations of other trigonometric graphs use the activity below to remind yourself what the graphs of trigonometric functions with vertical translations look like. © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES 25 Graphs of trigonometric functions 2 Using the equation y = a sin bx + c practice drawing graphs changing the values of a, b and c and the type of trigonometric function (sin, cos and tan). Here are some examples, notice how the values of a, b and c change the graph of the function. .......................................... Example Problem: Here is a sketch of the graph of y = a sin bx◦ + c, 0 ≤ x ≤ 360. © H ERIOT-WATT U NIVERSITY Go online 26 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES What are the values of a, b and c? Solution: Step 1: The graph goes between a maximum of 6 and a minimum of -4 so the amplitude = 6 - (-4) / = 5, hence a = 5. 2 Step 2: There are two complete waves in 360 ◦ so b = 2. (Alternatively since one wave = 180 ◦ , b = 360 ÷ 180 = 2). Step 3: So far we think the equation is y = 5 sin 2x ◦ . The graph of y = 5 sin 2x ◦ has a maximum of 5 and a minimum of -5 but the graph in the problem has a maximum of 6 and a minimum of -4. The graph has been moved up by 1 unit so c = 1. Hence the equation of the graph is y = 5sin 2x ◦ + 1 .......................................... Identifying trigonometric functions exercise 2 Go online Q24: Identify the trigonometric function, y = a sin bx◦ + c, 0 ≤ x ≤ 360, shown in the graph. .......................................... © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Q25: Identify the trigonometric function, y = a cos bx◦ + c, 0 ≤ x ≤ 360, shown in the graph. .......................................... Q26: Identify the trigonometric function, y = a cos bx◦ + c, 0 ≤ x ≤ 360, shown in the graph. .......................................... Q27: Identify the trigonometric function, y = a sin bx◦ + c, 0 ≤ x ≤ 360, shown in the graph. .......................................... Before we identify the equations of other trigonometric graphs use the activity below to remind yourself what the graphs of trigonometric functions with horizontal translations look like. © H ERIOT-WATT U NIVERSITY 27 28 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Graphs of trigonometric functions 3 Go online Using the equation y = a sin(x + b) + c practice drawing graphs changing the values of a, b and c and the type of trigonometric function (sin, cos and tan). Here are some examples, notice how the values of a, b and c change the graph of the function. .......................................... © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Examples 1. Problem: Here is a sketch of the graph of y = a sin (x + b◦ ) + c, 0 ≤ x ≤ 360. What are the values of a, b and c? Solution: Step 1: The graph goes between a maximum of 3 and a minimum of 1 so the amplitude = 3 - 1 / = 1, hence a = 1. 2 Step 2: There is one complete wave in 360 ◦ but the graph of y = sin x usually starts at the origin and this one does not. It starts at -30◦ . The graph has been moved left by 30 ◦ , hence b = 30. Step 3: So far we think the equation is y = 1 sin (x + 30 ◦ ). The graph of y = 1 sin (x + 30◦ ) has a maximum of 1 and a minimum of -1 but the graph in the problem has a maximum of 3 and a minimum of 1. The graph has been moved up by 2 units so c = 2. Hence the equation of the graph is y = sin(x + 30 ◦ ) + 2 .......................................... 2. Problem: Here is a sketch of the graph of y = a cos (x − b◦ ) + c, 0 ≤ x ≤ 360. What are the values of a, b and c? © H ERIOT-WATT U NIVERSITY 29 30 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Solution: Step 1: The graph goes between a maximum of 1 and a minimum of -3 so the amplitude = 1 - (-3) / = 2, hence a = 2. 2 Step 2: There is one complete wave in 360 ◦ but the graph of y = cos x usually starts at its maximum and this one does not. You should be able to see where the graph is at its maximum i.e. 60 ◦ . The graph has been moved right by 60 ◦ , hence b = 60. Step 3: So far we think the equation is y = 2 cos (x − 60 ◦ ). The graph of y = 2 cos (x − 60◦ ) has a maximum of 2 and a minimum of -2 but the graph in the problem has a maximum of 1 and a minimum of -3. The graph has been moved down by 1 unit so c = − 1. Hence the equation of the graph is y = 2cos(x − 60 ◦ ) − 1. .......................................... Identifying trigonometric functions exercise 3 Go online Q28: Identify the trigonometric function, y = a sin (x◦ + b) + c, 0 ≤ x ≤ 360, shown in the graph. .......................................... © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Q29: Identify the trigonometric function, y = a cos (x◦ + b) + c, 0 ≤ x ≤ 360, shown in the graph. .......................................... Q30: Identify the trigonometric function, y = a sin (x◦ + b) + c, 0 ≤ x ≤ 360, shown in the graph. .......................................... Q31: Identify the trigonometric function, y = a cos (x◦ + b) + c, 0 ≤ x ≤ 360, shown in the graph. .......................................... © H ERIOT-WATT U NIVERSITY 31 32 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES 1.3 Solving trigonometric equations Solving trigonometric equations Go online © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES .......................................... Key point To summarise, when solving a trig equation there are two solutions for 0 ≤ x ≤ 360◦ . Using the inverse trig functions on your calculator, you will get the acute angle a then: • for a sin equation the second solution is 180 − a; • for a tan equation the second solution is 180 + a. • for a cos equation the second solution is 360 − a; This is often memorised by the aid of a diagram. © H ERIOT-WATT U NIVERSITY 33 34 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Top tip There are many ways to remember the quadrant diagram e.g. • Going anti-clockwise round the quadrants this give ASTC . . . All Sinners Take Care. • Going anti-clockwise round the quadrants gives CAST. Key point You should draw a little quadrant chart every time you solve a trig equation to identify where you will find solutions. Examples 1. Problem: Solve the equation 3 sin x ◦ = 2, 0 ≤ x ≤ 360◦ Solution: Step 1: Rearrange the equation 3 sin x ◦ = 2 ⇒ sin x◦ = 2 3 Step 2: Draw a quadrant chart and tick the quadrants where sin has solutions, i.e. all and sin. Step 3: -1 Use your calculator◦ to find the first solution [using the sin button] ⇒ −1 2 = 41 · 8 sin 3 x = Step 4: Find the second solution [for sin use 180 − a] ⇒ x = 180 − 41 · 8 = 138 · 2◦ .......................................... 2. Problem: Solve the equation 5 cos x ◦ = 4, 0 ≤ x ≤ 360◦ . Solution: Step 1: Rearrange the equation 5 cos x ◦ = 4 ⇒ cos x◦ = 4 / 5. Step 2: Draw a quadrant chart and tick the quadrants where cos has solutions, i.e. all and cos. © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Step 3: -1 Use your 4 calculator◦ to find the first solution [using the cos button] ⇒ −1 = 36 · 9 cos 5 35 x = Step 4: Find the second solution [for cos use 360 − a] ⇒ x = 360 − 36 · 9 = 323 · 1◦ .......................................... 3. Problem: Solve the equation 3 tan x ◦ = 5, 0 ≤ x ≤ 360◦ Solution: Step 1: Rearrange the equation 3 tan x ◦ = 5 ⇒ tan x◦ = 5 3 Step 2: Draw a quadrant chart and tick the quadrants where tan has solutions, i.e. all and tan. Step 3: -1 Use your 5 calculator◦ to find the first solution [using the tan button] ⇒ −1 = 59 · 0 tan 3 x = Step 4: Find the second solution [tan so use 180 + a] ⇒ ⇒ x = 180 + 59 · 0 = 239 · 0 ◦ .......................................... Solving trigonometric equations practice (1) Q32: Solve the equation 4 sin x ◦ = 1, 0 ≤ x ≤ 360◦ .......................................... Q33: Solve the equation 5 cos x ◦ = 3, 0 ≤ x ≤ 360◦ .......................................... Q34: Solve the equation 7 tan x ◦ = 2, 0 ≤ x ≤ 360◦ .......................................... © H ERIOT-WATT U NIVERSITY Go online 36 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES So far we have looked for solutions where sin x, cos x and tan x equal a positive number. If sin x, cos x and tan x equal a negative number we must look for solutions in the opposite quadrants. If we take the graph of y = sin x you will see that the graph is positive between 0 ◦ and 180◦ and that corresponds to the All and Sin quadrants. The graph is negative between 180◦ and 360◦ and that corresponds to the T an and Cos quadrants. You know that cos x is positive in the All and Cos quadrants so it follows that cos x is negative in the Sin and T an quadrants. You know that tan x is positive in the All and T an quadrants so it follows that tan x is negative in the Sin and Cos quadrants. Example Problem: Solve the equation 5 sin x ◦ + 4 = 0, 0 ≤ x ≤ 360◦ Solution: Step 1: Rearrange the equation 5 sin x ◦ + 4 = 0 ⇒ sin x◦ = − 45 . Step 2: Draw a quadrant chart and tick the quadrants where sin has negative solutions, i.e. tan and cos. Step 3: Use your calculator to find the solution for sin x = sin-1 = 45 = 53 · 1◦ 4 5 [using the sin-1 button] ⇒ x = Step 4: Find the first solution [it’s the one in the tan quadrant so use 180 + a] ⇒ 180 + 53 · 1 = 233 · 1◦ x = © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Step 5: Find the second solution [it’s the one in the cos quadrant so use 360 − a] ⇒ 360 − 53 · 1 = 306 · 9◦ 37 x = .......................................... Key point It is extremely important to underline all your solutions when solving trig equations. Key point Always find the acute angle in the all quadrant first. This means never putting a negative value into your calculator to find the inverse trig value. Solving trigonometric equations practice (2) Q35: Solve the equation 2 sin x ◦ = − 1, 0 ≤ x ≤ 360◦ Go online .......................................... Solving trigonometric equations exercise Q36: Solve the equation 6 sin x = 1, 0 ≤ x ≤ 360 ◦ .......................................... Q37: Solve the equation 6 cos x = − 4, 0 ≤ x ≤ 360 ◦ .......................................... Q38: Solve the equation tan x = − 2, 0 ≤ x ≤ 360 ◦ .......................................... Q39: Solve the equation 8 sin x + 1 = 4, 0 ≤ x ≤ 360 ◦ .......................................... Q40: Solve the equation 9 cos x − 2 = 0, 0 ≤ x ≤ 360 ◦ .......................................... Q41: Solve the equation 7 tan x + 2 = 10, 0 ≤ x ≤ 360 ◦ .......................................... Q42: Solve the equation 5 sin x = − 4, 0 ≤ x ≤ 360 ◦ .......................................... Q43: Solve the equation 3 cos x − 2 = − 4, 0 ≤ x ≤ 360 ◦ .......................................... Q44: Solve the equation 2 tan x + 7 = 0, 0 ≤ x ≤ 360 ◦ .......................................... © H ERIOT-WATT U NIVERSITY Go online 38 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES 1.4 Exact trigonometric values Knowing the exact values of the trig ratios sin, cos and tan for certain angles can allow you to solve trig equations without a calculator. When x = 0 the values of sin x, cos x and tan x are easily seen from their graphs. Similarly when x = 90, 180, 270 and 360 the values of sin x, cos x and tan x can be identified from their graphs. The values are: sin x cos x tan x x = 0 0 1 0 x = 90◦ 1 0 undefined 180◦ 0 -1 0 x = 270◦ -1 0 undefined 360◦ 0 1 0 x = x = By constructing two triangles it will be possible to determine the exact values of some other common angles. Construction of 30◦ and 60◦ angles Sketch an equilateral triangle with side lengths of 2 units. Go online Draw in the perpendicular bisector from the apex to the base, focus on one of the right-angled triangles now formed. sides of the equilateral triangle are 2 units so the √ base of the right-angled triangle is 1 unit (base halved by bisector) and the height is 3 units (by Pythagoras). © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES height = = = √ √ 39 22 − 12 4 − 1 3 Read off the exact values for sin, cos and tan for the angles 30 ◦ and 60◦ using the SOH-CAH-TOA ratios. .......................................... Construction of 45◦ angles Sketch a square with side length 1 unit. Go online Draw in one of the diagonals, focus on one of the triangles now formed. The sides of the square are 1 unit√so the base and the height of the right-angled triangle are 1 unit and the hypotenuse is 2 units (by Pythagoras). 12 + 12 hypotenuse = √ = 1 + 1 √ 2 = © H ERIOT-WATT U NIVERSITY 40 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Read off the exact values for sin, cos and tan for the angles 30 ◦ and 60◦ using the SOH-CAH-TOA ratios. .......................................... The table below shows all the exact values between 0 ◦ and 90◦ . angle 0◦ sin x◦ 0 cos x◦ tan x◦ 1 0 30◦ 45◦ 60◦ 90◦ 1 2 √ 3 2 √1 3 √1 2 3 2 1 1 2 0 √1 2 1 √ √ 3 undefined Key point You will need to learn this table or remember how to construct the two special triangles. These exact values for trig. ratios can be extended to angles in all four quadrants. Recall the quadrant diagram. This can be remembered easily as All Students Talk Constantly and depicts the quadrants in which the various ratios are positive. The first quadrant has all ratios (A) positive. The second quadrant has sine (S) positive. The third quadrant has tangent (T) positive. The fourth quadrant has cosine (C) positive. To find an exact value for an angle greater than 90 ◦ , the angle is converted to the associated acute angle which lies between 0 and 90 in the All quadrant. You should recognise the diagram below from solving trig equations. © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES The following examples show how to find the associated acute angle which will be needed to determine the exact value. Examples 1. Problem: Find the associated acute angle for 120 ◦ . Solution: 120◦ is in the sin quadrant. The related angle is 60 ◦ because 180 − 60 = 120◦ . So the associated acute angle a = 60 ◦ . .......................................... 2. Problem: Find the associated acute angle for 300 ◦ . Solution: © H ERIOT-WATT U NIVERSITY 41 42 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES 300◦ is in the cos quadrant. The related angle is 60 ◦ because 360 − 60 = 300◦ . So the associated acute angle a = 60 ◦ . .......................................... 3. Problem: Find the associated acute angle for 210 ◦ . Solution: 210◦ is in the tan quadrant. The related angle is 30 ◦ because 180 + 30 = 210 ◦ . So the associated acute angle a = 30 ◦ . .......................................... Exact trigonometric values practice Q45: Find the associated acute angle for 315 ◦ . Go online .......................................... Q46: Find the associated acute angle for 150 ◦ . .......................................... Q47: Find the associated acute angle for 240 ◦ . .......................................... Exact values © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES .......................................... Example Problem: Find the exact value of cos 300◦ . Solution: 300◦ is in quadrant 4. Only cos is positive in the C quadrant so cos 300◦ is positive. The associated acute angle is found by 360 − a = 300 ◦ so a = 60◦ cos 300◦ = cos 60◦ 1 = 2 .......................................... © H ERIOT-WATT U NIVERSITY 43 44 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Example Problem: Find the exact value of sin 330◦ . Solution: 330◦ is in quadrant 4. Only cos is positive in the C quadrant so sin 330◦ is negative. The associated acute angle is found by 360 − a = 330 ◦ so a = 30◦ sin 330◦ = − sin 30◦ 1 = − 2 .......................................... Exact trigonometric values practice Go online Q48: Find the exact value of sin 150◦ . Is sin 150◦ positive or negative? a) Positive b) Negative .......................................... Q49: What is the associated acute angle? .......................................... Q50: What is the exact value of sin 150◦ ? a) b) c) √1 2 1 2 − 12 .......................................... Q51: Find the exact value of tan 150 ◦ . Is tan 150◦ positive or negative? a) Positive b) Negative .......................................... Q52: What is the associated acute angle? .......................................... Q53: What is the exact value of tan 150◦ ? a) √1 √3 b) 3 √ c) − 3 .......................................... © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES 45 Exact trigonometric values exercise Q54: Construct the two special triangles or recall the exact values to answer the following: a) sin 60 b) cos 30 c) tan 45 d) sin 270 e) cos180 f) tan 360 .......................................... Q55: Construct the two special triangles or recall the exact values to answer the following: a) sin 150 b) cos 330 c) tan 210 .......................................... Q56: Construct the two special triangles or recall the exact values to answer the following: a) sin 210 b) sin 300 c) sin 315 .......................................... Q57: Construct the two special triangles or recall the exact values to answer the following: a) sin 120 b) cos 300 c) tan 240 .......................................... Q58: Construct the two special triangles or recall the exact values to answer the following: a) cos 150 b) cos 315 c) cos 135 .......................................... © H ERIOT-WATT U NIVERSITY Go online 46 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Q59: Construct the two special triangles or recall the exact values to answer the following: a) sin 135 b) cos 315 c) tan 225 .......................................... Q60: Construct the two special triangles or recall the exact values to answer the following: a) tan 330 b) tan120 c) tan 315 .......................................... 1.5 Using trigonometric identities There are two trigonometric formulae to learn. From SOHCAHTOA we know that: sin x = O/ sin x H = A/ cos x H H O × = H A O = A = tan x 2 O H and cos x = A H 2 O 2 A + note: sin2 x means (sin x)2 H H O2 A2 + H2 H2 2 O + A2 H2 2 H by the Theorem of Pythagoras H2 1 2 sin x + cos x = = = = = © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Key point You will need to learn these trigonometric identities. 1. sin x cos x = tan x 2. sin2 x + cos2 x = 1 Note that rearranging gives: ⇒ sin2 x = 1 − cos2 x ⇒ cos2 x = 1 − sin2 x Examples 1. Problem: Simplify 4 sin2 x + 4cos2 x Solution: 4 sin2 x + 4 cos2 x = 4 sin2 x + cos2 x we know that sin2 x + cos2 x = 1 = 4 × 1 = 4 .......................................... 2. Problem: Solve 4 sin x − 5 cos x = 0 Solution: Divide each term by cos x. 4 sin x 5 cos x = cos0 x cos x − cos x sin x cos x now cos x = tan x, cos x = 1 and 0 cos x = 0 so, 4 tan x − 5 = 0 4 tan x = 5 5 tan x = 4 5 = 51 · 3◦ x = tan 4 x = 180 + 51 · 3 = 231 · 3◦ .......................................... −1 3. Problem: Prove that sin A sin2 x + sin A cos2 x = sinA Solution: The aim is to start with the left hand side of the equation and end up with the right hand side of the equation. © H ERIOT-WATT U NIVERSITY 47 48 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES LHS = sin A sin2 x + sin A cos2 x = sin A sin2 x + cos2 x = sin A × 1 = sin A = RHS This is called a mathematical proof. .......................................... Using trigonometric identities practice Q61: Prove that tan a sin a = Go online 1 − cos2 a cos a .......................................... Using trigonometric identities exercise Q62: Using sin2 x + cos2 x = 1 and tan x = Go online sin x cos x . a) What is sin2 x equal to? b) What is cos2 x equal to? c) What is sin x equal to? d) What is cos x equal to? .......................................... Q63: Simplify: a) 2 cos2 x + 2 sin2 x b) 3 sin A 3 cos A .......................................... Q64: Simplify: a) (1 + cos x) (1 − cos x) b) 2 − 2 sin2 A .......................................... Q65: Prove that: a) (sin X + cos X)2 = 1 + 2 sin X cos X b) (sin x − cos x)2 + (sin x + cos x)2 = 2 c) cos2 A − sin2 A = 2 cos2 A − 1 .......................................... © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES 1.6 Learning Points Sketching and identifying trigonometric graphs • The graph of y = sin x looks like this: • The graph of y = cos x looks like this: • The graph of y = tan x looks like this: • The graph of y = • • a sin bx has: ◦ a maximum of a; ◦ a minimum of −a; ◦ b complete waves in 360 ◦ . The graph of y = a cos bx + ◦ a maximum of a + c; ◦ a minimum of −a + c; ◦ b complete waves in 360 ◦ . The graph of y = a sin(x + ◦ a maximum of a + c; ◦ a minimum of −a + c; © H ERIOT-WATT U NIVERSITY c has: b) + c has: 49 50 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES ◦ been moved left or right horizontally by b ◦ . • The amplitude of a graph is half of the distance between the maximum and minimum values of the graph. • The period of a graph is the distance along the x-axis over which the graph completes one full wave pattern. • When identifying a trig graph: ◦ determine whether it is sin, cos or tan from its shape; ◦ identify the maximum and minimum values; ◦ determine the amplitude; ◦ identify the number of complete waves in 360 ◦ ; ◦ determine any vertical translation up or down; ◦ determine any horizontal translation left or right. Solving trigonometric equations • Rearrange the equation to sin x ◦ = or tan x◦ = or cos x◦ = • Draw a quadrant chart and tick the quadrants where the function has solutions. • Use your calculator to find the acute angle • • ◦ Use the sin-1 or tan-1 or cos-1 button. ◦ Never enter a negative to find the inverse trig value. Find the other solution(s): ◦ for the sin quadrant use 180 − a; ◦ for the tan quadrant use 180 + a; ◦ for the cos quadrant use 360 − a; Underline your solutions. Exact trigonometric values • Learn the table of exact values between 0 ◦ and 90◦ or remember how to construct the two special triangles. angle • 0◦ sin x◦ 0 cos x◦ 1 tan x◦ 0 30◦ 45◦ 60◦ 90◦ 1 2 √ 3 2 √1 2 3 2 1 1 2 0 √1 3 √1 2 1 √ √ 3 undefined Or learn how to draw the 2 special triangles. © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES • • Trigonometric identities • sin2 x + cos2 x = 1 • tan x = sin x cos x © H ERIOT-WATT U NIVERSITY 51 52 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES 1.7 End of topic test End of topic 21 test Sketching Trigonometric Graphs Go online Q66: Sketch the graph of y = cos 2x ◦ , for 0 ≤ x ≤ 360◦ .......................................... Q67: Sketch the graph of y = sin 3x ◦ , for 0 ≤ x ≤ 360◦ .......................................... Q68: Sketch the graph of y = tan 3x ◦ , for 0 ≤ x ≤ 360◦ .......................................... Q69: Sketch the graph of y = 4 sin 2x ◦ , for 0 ≤ x ≤ 360◦ .......................................... Q70: Sketch the graph of y = 4 tan 3x ◦ , for 0 ≤ x ≤ 360◦ .......................................... Q71: Sketch the graph of y = 3 cos 4x ◦ , for 0 ≤ x ≤ 360◦ .......................................... Q72: Sketch the graph of y = 3 sin 2x ◦ + 4, for 0 ≤ x ≤ 360◦ .......................................... Q73: Sketch the graph of y = 1 cos 4x ◦ − 3, for 0 ≤ x ≤ 360◦ .......................................... Q74: Sketch the graph of y = 1 tan 3x ◦ − 4, for 0 ≤ x ≤ 360◦ .......................................... Q75: Sketch the graph of y = 3 sin (x ◦ − 30), for 0 ≤ x ≤ 360◦ .......................................... Q76: Sketch the graph of y = − 3 cos (x ◦ + 90), for 0 ≤ x ≤ 360◦ .......................................... Q77: Sketch the graph of y = − tan (x ◦ − 60), for 0 ≤ x ≤ 360◦ .......................................... Q78: Sketch the graph of y = 2 tan (x ◦ + 90) + 2, for 0 ≤ x ≤ 360◦ .......................................... Q79: Sketch the graph of y = 3 cos (x ◦ + 15) − 1, for 0 ≤ x ≤ 360◦ .......................................... Q80: Sketch the graph of y = sin (x ◦ − 90) + 2, for 0 ≤ x ≤ 360◦ .......................................... © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Identifying Trigonometric Graphs Q81: a) Identify the trigonometric function, y = sin ax◦ , 0 ≤ x ≤ 360, shown in the graph above. b) What is the period? .......................................... Q82: a) Identify the trigonometric function, y = cos ax◦ , 0 ≤ x ≤ 360, shown in the graph above. b) What is the period? .......................................... © H ERIOT-WATT U NIVERSITY 53 54 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Q83: a) Identify the trigonometric function, y = a cos bx◦ , 0 ≤ x ≤ 360, shown in the graph above. b) What is the amplitude? .......................................... Q84: a) Identify the trigonometric function, y = a sin bx◦ , 0 ≤ x ≤ 360, shown in the graph above. b) What is the amplitude? .......................................... © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Q85: a) Identify the trigonometric function, y = a cos bx◦ + c, 0 ≤ x ≤ 360, shown in the graph above. b) What is the amplitude? .......................................... Q86: a) Identify the trigonometric function, y = a sin bx◦ + c, 0 ≤ x ≤ 360, shown in the graph above. b) What is the amplitude? .......................................... © H ERIOT-WATT U NIVERSITY 55 56 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Q87: a) Identify the trigonometric function, y = a tan bx◦ + c, 0 ≤ x ≤ 360, shown in the graph above. b) What is the amplitude? .......................................... Q88: Identify the trigonometric function, y = a cos (x◦ + b), 0 ≤ x ≤ 360, shown in the graph. .......................................... Q89: Identify the trigonometric function, y = a tan (x◦ + b), 0 ≤ x ≤ 360, shown in the graph. © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES .......................................... Q90: Identify the trigonometric function, y = a sin (x◦ + b), 0 ≤ x ≤ 360, shown in the graph. .......................................... Q91: Identify the trigonometric function, y = a tan (x◦ + b) + c, 0 ≤ x ≤ 360, shown in the graph. .......................................... Q92: Identify the trigonometric function, y = a sin (x◦ + b) + c, 0 ≤ x ≤ 360, shown in the graph. © H ERIOT-WATT U NIVERSITY 57 58 TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES .......................................... Q93: Identify the trigonometric function, y = a cos (x◦ + b) + c, 0 ≤ x ≤ 360, shown in the graph. .......................................... Solving Trigonometric Graphs Q94: Solve sin x = 0 · 18 .......................................... Q95: Solve 5 cos x = 2 .......................................... Q96: Solve 6 tan x − 2 = 0 .......................................... Q97: Solve 4 sin x = − 1 .......................................... Q98: Solve 7 cos x + 5 = 0 .......................................... Exact Trigonometric Graphs Give the exact values the following. Q99: sin 30◦ .......................................... Q100: cos 45◦ .......................................... Q101: tan 60◦ .......................................... © H ERIOT-WATT U NIVERSITY TOPIC 1. TRIGONOMETRIC EQUATIONS, GRAPHS AND IDENTITIES Q102: sin 135◦ .......................................... Q103: tan 330◦ .......................................... Q104: cos 240◦ .......................................... Trigonometric Identities Q105: Simplify 2cos2 Y + 2sin2 Y .......................................... Q106: Simplify sin3 A cos3 A .......................................... Q107: Solve 4 cos x + 5 sin x = 0 .......................................... Q108: Prove that (3cos x + 2sin x)2 + (2cos x − 3sin x)2 = 13 .......................................... .......................................... © H ERIOT-WATT U NIVERSITY 59 60 GLOSSARY Glossary perpendicular a line is perpendicular to another if they meet at right angles (90 degrees) perpendicular bisector a perpendicular bisector passes through the midpoint of the line it is perpendicular to © H ERIOT-WATT U NIVERSITY ANSWERS: TOPIC 21 Answers to questions and activities 21 Trigonometric equations, graphs and identities Graphs of trigonometric functions practice (page 5) Q1: Step 1: Start by plotting the important points from the sin curve on your graph. It starts at the origin and increases to the maximum value. Step 2: Sketch a smooth curve and label it. Q2: Step 1: Start by plotting the important points from the cos curve on your graph. It starts at the maximum value and decreases. © H ERIOT-WATT U NIVERSITY 61 62 ANSWERS: TOPIC 21 Step 2: Sketch a smooth curve and label it. Q3: Step 1: Start by plotting the important points on the tan curve and the dotted lines where the graph is undefined. It starts at the origin and increases but remember the tan curve is not continuous. © H ERIOT-WATT U NIVERSITY ANSWERS: TOPIC 21 Step 2: Sketch smooth curves and a label. Sketching trigonometric functions exercise 1 (page 9) Q4: © H ERIOT-WATT U NIVERSITY 63 64 ANSWERS: TOPIC 21 Q5: Q6: Q7: © H ERIOT-WATT U NIVERSITY ANSWERS: TOPIC 21 Q8: Sketching trigonometric functions exercise 2 (page 15) Q9: Q10: © H ERIOT-WATT U NIVERSITY 65 66 ANSWERS: TOPIC 21 Q11: Q12: Q13: © H ERIOT-WATT U NIVERSITY ANSWERS: TOPIC 21 Sketching trigonometric functions exercise 3 (page 21) Q14: Q15: Q16: © H ERIOT-WATT U NIVERSITY 67 68 ANSWERS: TOPIC 21 Q17: Q18: Q19: Identifying trigonometric functions exercise 1 (page 23) Q20: y = -4 sin 3·5x Q21: y = cos 2·5x Q22: y = 5 sin 0·5x © H ERIOT-WATT U NIVERSITY ANSWERS: TOPIC 21 Q23: y = -6 cos 4x Identifying trigonometric functions exercise 2 (page 26) Q24: y = 3 sin 2·5x - 3 Q25: y = 2 cos 2·5x - 4 Q26: y = -2 cos 0·5x - 2 Q27: y = -2 sin 3·5 + 4 Identifying trigonometric functions exercise 3 (page 30) Q28: y = -4 sin(x + 45) -1 Q29: y = -cos(x + 60) + 2 Q30: y = -2 sin(x + 45) + 3 Q31: y = 4 cos(x - 30) + 1 Solving trigonometric equations practice (1) (page 35) Q32: 4 sin x◦ = 1 1 sin x◦ = 4 1 = 14 · 5◦ x = sin 4 x◦ = 180 − 14 · 5 = 165 · 5◦ ◦ −1 Q33: 5 cos x◦ = 3 3 cos x◦ = 5 © H ERIOT-WATT U NIVERSITY 69 70 ANSWERS: TOPIC 21 3 = 53 · 1◦ 5 = 360 − 53 · 1 = 306 · 9◦ x◦ = cos−1 x◦ Q34: 7 tan x◦ = 2 2 tan x◦ = 7 x ◦ = tan −1 2 = 15 · 9◦ 7 x◦ = 180 + 15 · 9 = 195 · 9◦ Solving trigonometric equations practice (2) (page 37) Q35: 2 sin x◦ = −1 1 sin x◦ = − 2 When sin x = 12 , x = sin−1 12 = solutions in the tan and cos quadrants so, 30◦ but this is not a solution. We need x = 180 + 30 = 210◦ x = 360 − 30 = 330◦ Solving trigonometric equations exercise (page 37) Q36: x = 9 · 6◦ and x = 170 · 4◦ Q37: x = 131 · 8◦ and x = 228 · 2◦ Q38: x = 296 · 6◦ and x = 116 · 6◦ Q39: Steps: • Re-arrange the equation. © H ERIOT-WATT U NIVERSITY ANSWERS: TOPIC 21 • sin x = ? 3 /8 • Use this answer to find your solution remembering to press the sin -1 button. Answer: x = 22 · 0◦ and x = 158 · 0◦ Q40: Steps: • Re-arrange the equation. • cos x =? 2 /9 • Use this answer to find your solution remembering to press the cos -1 button. Answer: x = 77 · 2◦ and x = 282 · 8◦ Q41: Steps: • Re-arrange the equation. • tan x = ? 8 /7 • Use this answer to find your solution remembering to press the tan -1 button. Answer: x = 48 · 8◦ and x = 228 · 8◦ Q42: x = 306 · 8◦ and x = 233 · 1◦ Steps: • Re-arrange the equation. • sin x = ? −4 /5 • sin-1 (4 /5 ) = ? 53·1 • Remember to identify the quadrants where sin x is negative and use this answer to find your solutions. Answer: Q43: Steps: • Re-arrange the equation. • cos x = ? −2 /3 • cos-1 (2 /3 ) = ? 41·8 • Remember to identify the quadrants where cos x is negative and use this answer to find your solutions. Answer: x = 318 · 2◦ and x = 41 · 8◦ Q44: Steps: • Re-arrange the equation. • tan x = ? −7 /2 © H ERIOT-WATT U NIVERSITY 71 72 ANSWERS: TOPIC 21 • tan-1 (7 /2 ) = ? 74·1 • Remember to identify the quadrants where tan x is negative and use this answer to find your solutions. Answer: x = 285 · 9◦ and x = 105 · 9◦ Exact trigonometric values practice (page 42) Q45: 45 ◦ Q46: 30 ◦ Q47: 60 ◦ Exact trigonometric values practice (page 44) Q48: a) Positive Q49: 30 ◦ Q50: b) 1 2 Q51: b) Negative Q52: 30 ◦ √ Q53: c) − 3 Exact trigonometric values exercise (page 45) Q54: √ a) b) 3 2 √ 3 2 c) 1 d) -1 e) -1 f) 0 Q55: a) 1/2 √ b) c) 3 2 √1 3 Q56: a) -1/2 © H ERIOT-WATT U NIVERSITY ANSWERS: TOPIC 21 √ 3 2 − √12 b) − c) Q57: √ 3 2 a) b) 1/2 √ c) 3 Q58: √ 3 2 − √12 − √12 a) − b) c) Q59: a) b) √1 2 1 √ 2 c) 1 Q60: a) − √13 √ b) − 3 c) -1 Using trigonometric identities practice (page 48) Q61: RHS = tan a sin a sin a × sin a = cos a sin2 a if sin2 x + cos2 x = 1 then sin2 x = 1 − cos2 x = cos a 1 − cos2 a = cos a = LHS Using trigonometric identities exercise (page 48) Q62: a) 1 − sin2 x b) 1 − cos2 x © H ERIOT-WATT U NIVERSITY 73 74 ANSWERS: TOPIC 21 c) cos x tan x d) sin x tan x Q63: a) 2 b) tan A Q64: a) sin2 x b) 2 cos A Q65: a) RHS = (sin X + cos X)2 = sin X 2 + 2 sin X cos X + cos X 2 = sin X 2 + cos X 2 + 2 sin X cos X = 1 + 2 sin X cos X = LHS b) RHS = (sin x − cos x)2 + (sin x + cos x)2 = sin x2 − 2 sin x cos x + cos x2 + sin x2 + 2 sin x cos x + cos x2 = sin x2 + cos x2 + sin x2 + cos x2 − 2 sin x cos x + 2 sin x cos x = 1 + 1 = 2 = LHS c) RHS = cos2 A − sin2 A = cos2 A − 1 − cos2 A = cos2 A − 1 + cos2 A = 2cos2 A − 1 = LHS © H ERIOT-WATT U NIVERSITY ANSWERS: TOPIC 21 End of topic 21 test (page 52) Q66: Q67: Q68: © H ERIOT-WATT U NIVERSITY 75 76 ANSWERS: TOPIC 21 Q69: Q70: Q71: © H ERIOT-WATT U NIVERSITY ANSWERS: TOPIC 21 Q72: Q73: Q74: © H ERIOT-WATT U NIVERSITY 77 78 ANSWERS: TOPIC 21 Q75: Q76: Q77: © H ERIOT-WATT U NIVERSITY ANSWERS: TOPIC 21 Q78: Q79: Q80: Q81: a) y = sin 1 · 5x◦ b) 1·5 Q82: a) y = cos 0 · 5x◦ © H ERIOT-WATT U NIVERSITY 79 80 ANSWERS: TOPIC 21 b) 0·5 Q83: a) y = 4 cos 3 · 5x◦ b) 4 Q84: a) y = 3 sin 3 · 5x◦ b) 3 Q85: a) y = 4 cos 3 · 5x◦ − 1 b) 4 Q86: a) y = − 2 sin 4x◦ − 2 b) -2 Q87: a) y = − 3 tan 2 · 5x◦ − 2 b) -3 Q88: y = − 4 cos (x◦ + 60) Q89: y = 4 tan (x◦ − 30) Q90: y = 3 sin (x◦ + 15) Q91: y = − 2 tan (x◦ − 30) + 2 Q92: y = 4 sin (x◦ − 15) − 1 Q93: y = − 2 cos (x◦ − 30) − 2 Q94: x = 10 · 4 and x = 169 · 6 Q95: Hint • Put this in your calculator to help you find 2 solutions remembering to use a quadrant chart. Answer: x = 66 · 4 and x = 293 · 6 Q96: Hint • Put this in your calculator to help you find 2 solutions remembering to use a quadrant chart. © H ERIOT-WATT U NIVERSITY ANSWERS: TOPIC 21 Answer: x = 18 · 4 and x = 198 · 4 Q97: Hint • Put the positive value in your calculator to help you find 2 solutions remembering to use a quadrant chart. Answer: x = 194 · 5 and x = 345 · 5 Q98: Hint • Put the positive value in your calculator to help you find 2 solutions remembering to use a quadrant chart. Answer: x = 135 · 6 and x = 224 · 4 Q99: 1/2 Q100: Q101: √1 2 √ 3 Q102: Steps: • 135◦ is in the sin quadrant, what is the associated acute angle (Hint: 180 - a)? 45 • Use this to help you find the exact value. Answer: − √13 Q103: Steps: • 330◦ is in the cos quadrant, what is the associated acute angle (Hint: 360 - a)? 30 • Use this to help you find the exact value. Answer: √1 2 Q104: Steps: • 120◦ is in the tan quadrant ,what is the associated acute angle (Hint: 180 + a)? 60 • Use this to help you find the exact value. Answer: −1/2 Q105: 2 cos2 Y + 2 sin2 Y = 2 cos2 Y + sin2 Y = 2 × 1 = 2 © H ERIOT-WATT U NIVERSITY 81 82 ANSWERS: TOPIC 21 Q106: sin A × sin A × sin A sin3 A = 3 cos A cos A × cos A × cos A = tan A × tan A × tan A = tan3 A Q107: 4 cos x + 5 sin x = 0 sin x 0 cos x + 5 = 4 cos x cos x cos x 4 + 5 tan x = 0 5 tan x = −4 4 tan x = − 5 4 When tan x = 5 , x = 38 · 7 So x = 180 − 38 · 7 = 141 · 3 and x = 360 − 38 · 7 = 321 · 3 Q108: RHS = (3 cos x + 2 sin x)2 + (2 cos x − 3 sin x)2 = 9 cos2 x + 12 cos x sin x + 4 sin2 x + 4 cos2 x − 12 cos x sin x + 9 sin2 x = 13 cos2 x + 13 sin2 x = 13 cos2 x + sin2 x = 13 × 1 = 13 = LHS © H ERIOT-WATT U NIVERSITY