Introduction To Trigonometric Functions

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Introduction toTrigonometric FunctionsJackie NicholasPeggy AdamsonMathematics Learning CentreUniversity of SydneyNSW 2006c 1998University of Sydney

AcknowledgementsA significant part of this manuscript has previously appeared in a version of this bookletpublished in 1986 by Peggy Adamson. In rewriting this booklet, I have relied a great dealon Peggy’s ideas and approach for Chapters 1, 2, 3, 4, 5 and 7. Chapter 6 appears in asimilar form in the booklet, Introduction to Differential Calculus, which was written byChristopher Thomas.In her original booklet, Peggy acknowledged the contributions made by Mary Barnes andSue Gordon. I would like to extend this list and thank Collin Phillips for his hours ofdiscussion and suggestions.Jackie NicholasSeptember 1998

Contents1 Introduction11.1How to use this booklet . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 Angles and Angular Measure22.1Converting from radians to degrees and degrees to radians . . . . . . . . .32.2Real numbers as radians . . . . . . . . . . . . . . . . . . . . . . . . . . . .42.2.15Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Trigonometric Ratios in a Right Angled Triangle3.13.2Definition of sine, cosine and tangent . . . . . . . . . . . . . . . . . . . . .63.1.1Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7Some special trigonometric ratios . . . . . . . . . . . . . . . . . . . . . . .74 The Trigonometric Functions4.14.284.1.1Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9The sine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10The tangent function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3.14.48The cosine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.14.3Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Extending the domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4.1Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Graphs of Trigonometric Functions5.1Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Changing the mean level . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3.15.4Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Changing the period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2.15.314Changing the amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.1.15.26Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Changing the phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4.1Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20i

Mathematics Learning Centre, University of Sydneyii6 Derivatives of Trigonometric Functions216.1The calculus of trigonometric functions . . . . . . . . . . . . . . . . . . . . 216.1.1Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 A Brief Look at Inverse Trigonometric Functions7.123Definition of the inverse cosine function . . . . . . . . . . . . . . . . . . . . 247.1.1Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 Solutions to Exercises26

1Mathematics Learning Centre, University of Sydney1IntroductionYou have probably met the trigonometric ratios cosine, sine, and tangent in a right angledtriangle, and have used them to calculate the sides and angles of those triangles.In this booklet we review the definition of these trigonometric ratios and extend theconcept of cosine, sine and tangent. We define the cosine, sine and tangent as functionsof all real numbers. These trigonometric functions are extremely important in science,engineering and mathematics, and some familiarity with them will be assumed in mostfirst year university mathematics courses.In Chapter 2 we represent an angle as radian measure and convert degrees to radiansand radians to degrees. In Chapter 3 we review the definition of the trigonometric ratiosin a right angled triangle. In Chapter 4, we extend these ideas and define cosine, sineand tangent as functions of real numbers. In Chapter 5, we discuss the properties oftheir graphs. Chapter 6 looks at derivatives of these functions and assumes that youhave studied calculus before. If you haven’t done so, then skip Chapter 6 for now. Youmay find the Mathematics Learning Centre booklet: Introduction to Differential Calculususeful if you need to study calculus. Chapter 7 gives a brief look at inverse trigonometricfunctions.1.1How to use this bookletYou will not gain much by just reading this booklet. Mathematics is not a spectator sport!Rather, have pen and paper ready and try to work through the examples before readingtheir solutions. Do all the exercises. It is important that you try hard to complete theexercises, rather than refer to the solutions as soon as you are stuck.1.2ObjectivesBy the time you have completed this booklet you should: know what a radian is and know how to convert degrees to radians and radians todegrees; know how cos, sin and tan can be defined as ratios of the sides of a right angledtriangle; know how to find the cos, sin and tan of π6 , know how cos, sin and tan functions are defined for all real numbers; be able to sketch the graph of certain trigonometric functions; know how to differentiate the cos, sin and tan functions; understand the definition of the inverse function f 1 (x) cos 1 (x).π4and π2 ;

2Mathematics Learning Centre, University of Sydney2Angles and Angular MeasureAn angle can be thought of as the amount of rotation required to take one straight lineto another line with a common point. Angles are often labelled with Greek letters, forexample θ. Sometimes an arrow is used to indicate the direction of the rotation. If thearrow points in an anticlockwise direction, the angle is positive. If it points clockwise, theangle is negative.BOAAngles can be measured in degrees or radians. Measurement in degrees is based ondividing the circumference of the circle into 360 equal parts. You are probably familiarwith this method of measurement.o3 601 80o.A complete revolution is360 .A straight angle is 180 .90oA right angle is 90 .Fractions of a degree are expressed in minutes ( ) and seconds ( ). There are sixty secondsin one minute, and sixty minutes in one degree. So an angle of 31 17 can be expressedas 31 17 31.28 .60The radian is a natural unit for measuring angles. We use radian measure in calculusbecause it makes the derivatives of trigonometric functions simple. You should try to getused to thinking in radians rather than degrees.To measure an angle in radians, construct a unit circle(radius 1) with centre at the vertex of the angle. Theradian measure of an angle AOB is defined to be the lengthof the circular arc AB around the circumference.B1OAThis definition can be used to find the number of radians corresponding to one completerevolution.

3Mathematics Learning Centre, University of SydneyIn a complete revolution, A moves anticlockwise aroundthe whole circumference of the unit circle, a distance of2π. So a complete revolution is measured as 2π radians.That is, 2π radians corresponds to 360 .1OAFractions of a revolution correspond to angles which are fractions of 2π.142.1revolution 90 or π2 radians13revolution 120 or 2πradians3 16 revolution 60 or π3 radiansConverting from radians to degrees and degrees to radiansSince 2π radians is equal to 360 π radians 180 ,180 1 radian π 57.3 ,y radians y 180 ,πand similarly1 πradians,180 0.017,y y πradians.180Your calculator has a key that enters the approximate value of π.

4Mathematics Learning Centre, University of SydneyIf you are going to do calculus, it is important to get used to thinking in terms of radianmeasure. In particular, think of:180 as π radians,90 asπradians,260 asπradians,345 asπradians,430 asπradians.6You should make sure you are really familiar with these.2.2Real numbers as radiansAny real number can be thought of as a radian measure if we express the number as amultiple of 2π.B1π5π 2π (1 ) 2π corresponds to242the arc length of 1 14 revolutions of the unit circle goinganticlockwise from A to B.For example,OASimilarly,27 4.297 2π 4 2π 0.297 2πcorresponds to an arc length of 4.297 revolutions of the unit circle going anticlockwise.

5Mathematics Learning Centre, University of SydneyWe can also think of negative numbers in terms of radians. Remember for negative radianswe measure arc length clockwise around the unit circle.For example, 16 2.546 2π 2 2π 0.546 2πcorresponds to the arc length of approximately 2.546revolutions of the unit circle going clockwise from Ato B.BOAWe are, in effect, wrapping the positive real number line anticlockwise around the unitcircle and the negative real number line clockwise around the unit circle, starting in eachcase with 0 at A, (1, 0).By doing so we are associating each and every real number with exactly one point on theunit circle. Real numbers that have a difference of 2π (or a multiple of 2π) correspond tothe same point on the unit circle. Using one of our previous examples, 5πcorresponds to2πas they differ by a multiple of 2π.22.2.1ExerciseWrite the following in both degrees and radians and represent them on a diagram.a. 30 d.3π4g. 270 b. 1c. 120 e. 2f.h. 1i. π24π3Note that we do not indicate the units when we are talking about radians.In the rest of this booklet, we will be using radian measure only. You’ll needto make sure that your calculator is in radian mode.

6Mathematics Learning Centre, University of Sydney3Trigonometric Ratios in a Right Angled TriangleIf you have met trigonometry before, you probably learned definitions of sin θ, cos θ andtan θ which were expressed as ratios of the sides of a right angled triangle.These definitions are repeated here, just to remind you, but we shall go on, in the nextsection, to give a much more useful definition.3.1Definition of sine, cosine and tangentIn a right angled triangle, the side opposite to theright angle is called the hypotenuse. If we choose oneof the other angles and label it θ, the other sides areoften called opposite (the side opposite to θ) and adjacent (the side next to θ).HypotenuseOppositeθAdjacentFor a given θ, there is a whole family of right angled triangles, that are triangles of differentsizes but are the same shape.θθθFor each of the triangles above, the ratios of corresponding sides have the same values.adjacent has the same value for each triangle. This ratio is given a specialThe ratio hypotenusename, the cosine of θ or cos θ.opposite has the same value for each triangle. This ratio is the sine of θ orThe ratio hypotenusesin θ.opposite takes the same value for each triangle. This ratio is called the tangentThe ratio adjacentof θ or tan θ.Summarising,cos θ adjacent,hypotenusesin θ opposite,hypotenusetan θ opposite.adjacent

7Mathematics Learning Centre, University of SydneyThe values of these ratios can be found using a calculator. Remember, we are working inradians so your calculator must be in radian mode.3.1.1ExerciseUse your calculator to evaluate the following. Where appropriate, compare your answerswith the exact values for the special trigonometric ratios given in the next section.a. sin π6b. tan 1c. cos π3d. tan π4e. sin 1.5f. tan π3g. cos π6h. sin π33.2Some special trigonometric ratiosYou will need to be familiar with the trigonometric ratios of π6 , π3 and π4 .The ratios of π6 and π3 are found with the aid of an equilateral triangle ABC with sides oflength 2. BAC is bisected by AD, and ADC is a right angle.Pythagoras’theorem tells us that the length of AD 3. AACD π3 .DAC π6 .2C1π ,cos32 π3sin ,32 πtan3. 32 311 π3cos ,62π1sin ,62π1tan .63The ratios of π4 are found with the aid of an isoscelesright angled triangle XYZ with the two equal sides oflength 1.Xπ/4Pythagoras’ theorem tells us that the hypotenuse ofthe triangle has length 2.1π ,cos421π ,sin42πtan 1.4BD 2Zπ/411Y

8Mathematics Learning Centre, University of Sydney4The Trigonometric FunctionsThe definitions in the previous section apply to θ between 0 and π2 , since the angles in aright angle triangle can never be greater than π2 . The definitions given below are usefulin calculus, as they extend sin θ, cos θ and tan θ without restrictions on the value of θ.4.1The cosine functionLet’s begin with a definition of cos θ.Consider a circle of radius 1, with centre O at the origin ofthe (x, y) plane. Let A be the point on the circumferenceof the circle with coordinates (1, 0). OA is a radius of thecircle with length 1. Let P be a point on the circumferenceof the circle with coordinates (a, b). We can represent theangle between OA and OP, θ, by the arc length along theunit circle from A to P. This is the radian representationof θ.P(a,b)OθQAThe cosine of θ is defined to be the x coordinate of P.Let’s, for the moment, consider values of θ between 0 and π2 . The cosine of θ is writtencos θ, so in the diagram above, cos θ a. Notice that as θ increases from 0 to π2 , cos θdecreases from 1 to 0.For values of θ between 0 and π2 , this definition agrees with the definition of cos θ as theadjacent of the sides of a right angled triangle.ratio hypotenuseDraw PQ perpendicular OA. In OPQ, the hypotenuse OP has length 1, while OQ haslength a.adjacent a cos θ.The ratio hypotenuseThe definition of cos θ using the unit circle makes sense for all values of θ. For now, wewill consider values of θ between 0 and 2π.The x coordinate of P gives the value of cos θ. When θ π2 , P is on the y axis, and it’sx coordinate is zero. As θ increases beyond π2 , P moves around the circle into the secondquadrant and therefore it’s x coordinate will be negative. When θ π, the x coordinateis 1.PPPOθAcos θ is positiveθOcos π2 0Oθcos θ negativePOθcos π 1

9Mathematics Learning Centre, University of SydneyAs θ increases further, P moves around into the third quadrant and its x coordinateto 2π the x coordinate of Pincreases from 1 to 0. Finally as θ increases from 3π2increases from 0 to 1.OOOPOPPPcos 3π 02cos θ is negative4.1.1cos θ positivecos 2π 1Exercise1. Use the cosine (cos) key on your calculator to complete this table. (Make sure yourcalculator is in radian �44π33π25π37π42πcos θθcos θ2. Using this table plot the graph of y cos θ for values of θ ranging from 0 to 2π.4.2The sine functionThe sine of θ is defined using the same unit circle diagramthat we used to define the cosine.The sine of θ is defined to be the y coordinate ofP.P(a,b)OθQAThe sine of θ is written as sin θ, so in the diagram above, sin θ b.For values of θ between 0 and π2 , this definition agrees with the definition of sin θ as theopposite of sides of a right angled triangle.ratio hypotenuseIn the right angled triangle OQP, the hypotenuse OP has length 1 while PQ has length b.opposite The ratio hypotenuseb1 sin θ.This definition of sin θ using the unit circle extends to all values of θ. Here, we willconsider values of θ between 0 and 2π.

10Mathematics Learning Centre, University of SydneyAs P moves anticlockwise around the circle from A to B, θ increases from 0 to π2 . WhenP is at A, sin θ 0, and when P is at B, sin θ 1. So as θ increases from 0 to π2 , sin θincreases from 0 to 1. The largest value of sin θ is 1.As θ increases beyond π2 , sin θ decreases and equals zero when θ π. As θ increasesbeyond π, sin θ becomes negative.PPPOθθAOOsin π2 1sin θ is positiveOθPOsin θ positiveθsin π 0OOOPPP 1sin 3π2sin θ is negative4.2.1Psin θ negativesin 2π 0Exercise1. Use the sin key on your calculator to complete this table. Make sure your calculatoris in radian mode.θ0 0.2 0.4 0.6 0.811.2 1.4 1.6sin θθ2 2.4 2.8 3.2 3.6 4.0 4.6 5.4 6.2sin θ2. Plot the graph of the y sin θ using the table in the previous exercise.4.3The tangent functionWe can define the tangent of θ, written tan θ, in terms of sin θ and cos θ.tan θ sin θ.cos θUsing this definition we can work out tan θ for values of θ between 0 and 2π. You will

11Mathematics Learning Centre, University of Sydneybe asked to do this in Exercise 3.3. In particular, we know from this definition that tan θ.is not defined when cos θ 0. This occurs when θ π2 or θ 3π2oppositeWhen 0 θ π2 this definition agrees with the definition of tan θ as the ratio adjacentof the sides of a right angled triangle.As before, consider the unit circle with points O, A and Pas shown. Drop a perpendicular from the point P to OAwhich intersects OA at Q. As before P has coordinates(a, b) and Q coordinates (a, 0).P(a,b)oppositePQ (in triangle OPQ)adjacentOQb asin θ cos θ tan θ.OθQAIf you try to find tan π2 using your calculator, you will get an error message. Look at thedefinition. The tangent of π2 is not defined as cos π2 0. For values of θ near π2 , tan θ isvery large. Try putting some values in your calculator. (eg π2 1.570796. Try tan(1.57),tan(1.5707), tan(1.57079).)4.3.1Exercise1. Use the tan key on your calculator to complete this table. Make sure your calculatoris in radian mode.θ00.2 0.4 0.6 0.8 1.01.21.41.50 1.6522.4tan θθ2.8 3.2 3.6 4.0 4.4 4.6 4.65 4.78tan θ2. Use the table above to graph tan θ.5.05.66.0 6.28

12Mathematics Learning Centre, University of SydneyYour graph should look like this for values of θ between 0 and π.Notice that there is a vertical asymptoteat θ π2 . This is because tan θ is notdefined at θ π2 . You will find anothervertical asymptote at θ 3π. When θ 20 or π, tan θ 0. For θ greater than 0and less than π2 , tan θ is positive. Forvalues of θ greater than π2 and less thanπ, tan θ is negative.4.4Oπ /2πExtending the domainThe definitions of sine, cosine and tangent can be extended to all real values of θ in thefollowing way.π5π 2π corresponds to the arc length of 1 14 revolu22tions around the unit circle going anticlockwise from A toB.Since B has coordinates (0, 1) we can use the previousdefinitions to get:sin5π2BAO 1,cos 5π 0,2tan 5πis undefined.2Similarly, 16 sin( 16) cos( 16) tan( 16) 2.546 2π 2 2π 0.546 2π,sin( 0.546 2π)0.29, 0.96, 0.30.BOA

13Mathematics Learning Centre, University of Sydney4.4.1ExerciseEvaluate the following trig functions giving exact answers where you are able.1.sin 15π22.tan 13π63.cos 154.tan 14π35.sin 23π6NoticeThe values of sine and cosine functions repeat after every interval of length 2π. Sincethe real numbers x, x 2π, x 2π, x 4π, x 4π etc differ by a multiple of 2π, theycorrespond to the same point on the unit circle. So, sin x sin(x 2π) sin(x 2π) sin(x 4π) sin(x 4π) etc. We can see the effect of this in the functions below andwill discuss it further in the next chapter.1sin θθ 2π0 ππ2π 11cosθθ 2π0 ππ2π 1The tangent function repeats after every interval of length π.tanθ1θ 2π0 π 1π2π

14Mathematics Learning Centre, University of Sydney5Graphs of Trigonometric FunctionsIn this section we use our knowledge of the graphs y sin x and y cos x to sketch thegraphs of more complex trigonometric functions.1sin xx 2π0 ππ2ππ2π 11cosxx 2π0 π 1Let’s look first at some important features of these two graphs.The shape of each graph is repeated after every interval of length 2π.This makes sense when we think of the way we have defined sin and cos using the unitcircle.We say that these functions are periodic with period 2π.The sin and cos functions are the most famous examples of a class of functions calledperiodic functions.Functions with the property that f (x) f (x a) for all x are called periodicfunctions. Such a function is said to have period a.This means that the function repeats itself after every interval of length a.Note that you can have periodic

6.1.1 Exercise . 22 7 A Brief Look at Inverse Trigonometric Functions 23 7.1 Definition of the inverse cosine function . 24 7.1.1 Exercise . 25 8 Solutions to Exercises 26. Mathematics Learning Centre, University of Sydney 1 1 Introduction . example θ. Sometimes an arr

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