Lecture notes on Trigonometric functions

how trigonometric functions are used in real life, what makes a trigonometric function even or odd and what trigonometric function represents the graph
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Mathematics Learning Centre Introduction to Trigonometric Functions Peggy Adamson and Jackie Nicholas  c 1998 University of SydneyMathematics Learning Centre, University ofSydney 1 1Introduction Youhave probably met the trigonometric ratios cosine, sine, and tangent in a right angled triangle, 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 the concept of cosine, sine and tangent. We define the cosine, sine and tangent as functions of all real numbers. These trigonometric functions are extremely important in science, engineering and mathematics, and some familiarity with them will be assumed in most first year university mathematics courses. In Chapter 2 we represent an angle as radian measure and convert degrees to radians and radians to degrees. In Chapter 3 we review the definition of the trigonometric ratios in a right angled triangle. In Chapter 4, we extend these ideas and define cosine, sine and tangent as functions of real numbers. In Chapter 5, we discuss the properties of their graphs. Chapter 6 looks at derivatives of these functions and assumes that you have studied calculus before. If you haven’t done so, then skip Chapter 6 for now. You may find the Mathematics Learning Centre booklet: Introduction to Differential Calculus useful if you need to study calculus. Chapter 7 gives a brief look at inverse trigonometric functions. 1.1 How to use this booklet You 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 reading their solutions. Do all the exercises. It is important that you try hard to complete the exercises, rather than refer to the solutions as soon as you are stuck. 1.2 Objectives By the time you have completed this booklet you should: • know what a radian is and know how to convert degrees to radians and radians to degrees; • know how cos, sin and tan can be defined as ratios of the sides of a right angled triangle; π π π • know how to find the cos, sin and tan of , and ; 6 4 2 • 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; −1 −1 • understand the definition of the inverse function f (x)= cos (x).Mathematics Learning Centre, University ofSydney 2 2 Angles and Angular Measure An angle can be thought of as the amount of rotation required to take one straight line to another line with a common point. Angles are often labelled with Greek letters, for example θ. Sometimes an arrow is used to indicate the direction of the rotation. If the arrow points in an anticlockwise direction, the angle is positive. If it points clockwise, the angle is negative. B O A Angles can be measured in degrees or radians. Measurement in degrees is based on dividing the circumference of the circle into 360 equal parts. You are probably familiar with this method of measurement. o 360 o 180 . o 90 ◦ ◦ A complete revolution is A straight angle is 180.A right angle is 90 . ◦ 360 .   Fractions of a degree are expressed in minutes ( ) and seconds ( ). There are sixty seconds ◦  in one minute, and sixty minutes in one degree. So an angle of 31 17 can be expressed 17 ◦ as 31 + =31.28 . 60 The radian is a natural unit for measuring angles. We use radian measure in calculus because it makes the derivatives of trigonometric functions simple. You should try to get used to thinking in radians rather than degrees. B To measure an angle in radians, construct a unit circle 1 (radius 1) with centre at the vertex of the angle. The radian measure of an angle AOB is defined to be the length of the circular arc AB around the circumference. O A This definition can be used to find the number of radians corresponding to one complete revolution.Mathematics Learning Centre, University ofSydney 3 In a complete revolution, A moves anticlockwise around 1 the whole circumference of the unit circle, a distance of 2π.Soa complete revolution is measured as 2π radians. ◦ O That is, 2π radians corresponds to 360 . A Fractions of a revolution correspond to angles which are fractions of 2π. 1 1 1 ◦ ◦ ◦ revolution 90 revolution 120 − revolution−60 4 3 6 π 2π π or radians or radians or− radians 2 3 3 2.1 Converting from radians to degrees and degrees to radians ◦ Since 2π radians is equal to 360 ◦ π radians = 180 , ◦ 180 1 radian = π ◦ =57.3 , ◦ 180 y radians = y× , π and similarly π ◦ 1 = radians, 180 ≈ 0.017, π ◦ y = y× radians. 180 Your calculator has a key that enters the approximate value of π.Mathematics Learning Centre, University ofSydney 4 If you are going to do calculus, it is important to get used to thinking in terms of radian measure. In particular, think of: ◦ 180 as π radians, π ◦ 90 as radians, 2 π ◦ 60 as radians, 3 π ◦ 45 as radians, 4 π ◦ 30 as radians. 6 You should make sure you are really familiar with these. 2.2 Real numbers as radians Any real number can be thought of as a radian measure if we express the number as a multiple of 2π. B 5π 1 π For example, =2π× (1 + )=2π + corresponds to A 2 4 2 O 1 the arc length of 1 revolutions of the unit circle going 4 anticlockwise from A to B. Similarly, 27 ≈ 4.297× 2π =4× 2π+0.297× 2π corresponds to an arc length of 4.297 revolutions of the unit circle going anticlockwise.Mathematics Learning Centre, University ofSydney 5 We can also think of negative numbers in terms of radians. Remember for negative radians we measure arc length clockwise around the unit circle. For example, B −16≈−2.546× 2π =−2× 2π +−0.546× 2π A corresponds to the arc length of approximately 2.546 O revolutions of the unit circle going clockwise from A to B. We are, in effect, wrapping the positive real number line anticlockwise around the unit circle and the negative real number line clockwise around the unit circle, starting in each case with 0 at A, (1, 0). By doing so we are associating each and every real number with exactly one point on the unit circle. Real numbers that have a difference of 2π (or a multiple of 2π) correspond to 5π the same point on the unit circle. Using one of our previous examples, corresponds to 2 π as they differ by a multiple of 2π. 2 2.2.1 Exercise Write the following in both degrees and radians and represent them on a diagram. ◦ ◦ a. 30 b. 1 c. 120 3π 4π d. e. 2 f. 4 3 π ◦ g. 270 h. −1 i. − 2 Note 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 need to make sure that your calculator is in radian mode.Mathematics Learning Centre, University ofSydney 6 3Trigonometric Ratios in a Right Angled Triangle If you have met trigonometry before, you probably learned definitions of sinθ, cos θ and tanθ 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 next section, to give a much more useful definition. 3.1 Definition of sine, cosine and tangent In a right angled triangle, the side opposite to the right angle is called the hypotenuse.Ifwechoose one Hypotenuse of the other angles and label it θ, the other sides are Opposite often called opposite (the side opposite to θ) and ad- θ jacent (the side next to θ). Adjacent For a given θ, there is a whole family of right angled triangles, that are triangles of different sizes but are the same shape. θ θθ For each of the triangles above, the ratios of corresponding sides have the same values. adjacent The ratio has the same value for each triangle. This ratio is given a special hypotenuse name, the cosine of θ or cosθ. opposite The ratio has the same value for each triangle. This ratio is the sine of θ or hypotenuse sin θ. opposite The ratio takes the same value for each triangle. This ratio is called the tangent adjacent of θ or tanθ. Summarising, adjacent cos θ = , hypotenuse opposite sin θ = , hypotenuse opposite tanθ = . adjacentMathematics Learning Centre, University ofSydney 7 The values of these ratios can be found using a calculator. Remember, we are working in radians so your calculator must be in radian mode. 3.1.1 Exercise Use your calculator to evaluate the following. Where appropriate, compare your answers with the exact values for the special trigonometric ratios given in the next section. π π π a. sin b. tan 1 c. cos d. tan 6 3 4 π π π e. sin 1.5 f. tan g. cos h. sin 3 6 3 3.2 Some special trigonometric ratios π π π You will need to be familiar with the trigonometric ratios of , and . 6 3 4 π π The ratios of and are found with the aid of an equilateral triangle ABC with sides of 6 3 length 2.   BAC is bisected by AD, and ADC is a right angle. A Pythagoras’ theorem tells us that the length of AD = √ 3. π  ACD = . 2 2 3 3 √ π  DAC = . 6 11 C B D √ π 1 π 3 cos = , cos = , 3 2 6 2 √ π 3 π 1 sin = , sin = , 3 2 6 2 √ π π 1 tan = 3. √ tan = . 3 6 3 π The ratios of are found with the aid of an isosceles 4 right angled triangle XYZ with the two equal sides of length 1. X π/4 Pythagoras’ theorem tells us that the hypotenuse of √ the triangle has length 2. 2 √ 1 π 1 √ cos = , 4 2 π 1 π/4 1 sin = √ , Z Y 4 2 π tan =1. 4Mathematics Learning Centre, University ofSydney 8 4 The Trigonometric Functions π The definitions in the previous section apply to θ between 0 and , since the angles in a 2 π right angle triangle can never be greater than . The definitions given below are useful 2 in calculus, as they extend sinθ, cos θ and tanθ without restrictions on the value of θ. 4.1 The cosine function Let’s begin with a definition of cosθ. Consider a circle of radius 1, with centre O at the origin of P(a,b) the (x, y) plane. Let A be the point on the circumference of the circle with coordinates (1, 0). OA is a radius of the circle with length 1. Let P be a point on the circumference θ A O of the circle with coordinates (a, b). We can represent the Q angle between OA and OP, θ,by the arc length along the unit circle from A to P. This is the radian representation of θ. The cosine of θ is defined to be the x coordinate of P. π Let’s, for the moment, consider values of θ between0 and . The cosine of θ is written 2 π cos θ,soin the diagram above, cosθ = a. Notice that as θ increases from 0 to , cos θ 2 decreases from 1 to 0. π Forvalues of θ between 0 and , this definition agrees with the definition of cosθ as the 2 adjacent ratio of the sides of a right angled triangle. hypotenuse Draw PQ perpendicular OA. In OPQ, the hypotenuse OP has length 1, while OQ has length a. adjacent The ratio = a = cos θ. hypotenuse The definition of cosθ using the unit circle makes sense for all values of θ.For now, we will consider values of θ between 0 and 2π. π The x coordinate of P gives the value of cosθ. When θ = ,Pis on the y axis, and it’s 2 π x coordinate is zero. As θ increases beyond ,P moves around the circle into the second 2 quadrant and therefore it’s x coordinate will be negative. When θ = π, the x coordinate is−1. P P P θ θθθ A P O O O O π cos θ is positive cos =0 cos θ negative cosπ =−1 2Mathematics Learning Centre, University ofSydney 9 As θ increases further, P moves around into the third quadrant and its x coordinate 3π increases from −1to0. Finally as θ increases from to 2π the x coordinate of P 2 increases from 0 to 1. O O O P O P P P 3π cos θ is negative cos =0 cos θ positive cos 2π =1 2 4.1.1 Exercise 1. Use the cosine (cos) key on your calculator to complete this table. (Make sure your calculator is in radian mode.) π π π π 5π π 2π 3π θ 0 12 6 4 3 12 2 3 4 cos θ 5π 7π 5π 4π 3π 5π 7π θ π 2π 6 6 4 3 2 3 4 cos θ 2. Using this table plot the graph of y = cos θ for values of θ ranging from 0 to 2π. 4.2 The sine function The sine of θ is defined using the same unit circle diagram P(a,b) that we used to define the cosine. θ A O Q The sine of θ is defined to be the y coordinate of P. The sine of θ is written as sinθ,soin the diagram above, sinθ = b. π Forvalues of θ between0 and , this definition agrees with the definition of sinθ as the 2 opposite ratio of sides of a right angled triangle. hypotenuse In the right angled triangle OQP, the hypotenuse OP has length 1 while PQ has length b. opposite b The ratio = = sin θ. 1 hypotenuse This definition of sinθ using the unit circle extends to all values of θ. Here, we will consider values of θ between 0 and 2π.Mathematics Learning Centre, University ofSydney 10 π As P moves anticlockwise around the circle from A to B, θ increases from 0 to . When 2 π PisatA,sin θ =0, and when P is at B, sin θ =1. So as θ increases from 0 to , sin θ 2 increases from 0 to 1. The largest value of sinθ is 1. π As θ increases beyond , sin θ decreases and equals zero when θ = π.As θ increases 2 beyond π, sin θ becomes negative. P P P θ θθθ A P O O O O π sin θ is positive sin =1 sin θ positive sinπ =0 2 O O O P O P P P 3π sin θ is negative sin =−1 sin θ negative sin 2π =0 2 4.2.1 Exercise 1. Use the sin key on your calculator to complete this table. Make sure your calculator is in radian mode. θ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 sin θ θ 2 2.4 2.8 3.2 3.6 4.0 4.6 5.4 6.2 sin θ 2. Plot the graph of the y = sin θ using the table in the previous exercise. 4.3 The tangent function We can define the tangent of θ, written tanθ,in terms of sinθ and cosθ. sin θ tanθ = . cos θ Using this definition we can work out tanθ for values of θ between 0 and 2π.You willMathematics Learning Centre, University ofSydney 11 be asked to do this in Exercise 3.3. In particular, we know from this definition that tanθ π 3π is not defined when cos θ =0. This occurs when θ = or θ = . 2 2 opposite π When 0θ this definition agrees with the definition of tanθ as the ratio 2 adjacent of the sides of a right angled triangle. As before, consider the unit circle with points O, A and P as shown. Drop a perpendicular from the point P to OA P(a,b) which intersects OA at Q. As before P has coordinates (a, b) and Q coordinates (a, 0). θ A opposite PQ O Q = (in triangle OPQ) adjacent OQ b = a sin θ = cos θ = tanθ. π If you try to find tan using your calculator, you will get an error message. Look at the 2 π π π definition. The tangent of is not defined as cos =0.For values of θ near , tanθ is 2 2 2 π very large. Try putting some values in your calculator. (eg ≈ 1.570796. Try tan(1.57), 2 tan(1.5707), tan(1.57079).) 4.3.1 Exercise 1. Use the tan key on your calculator to complete this table. Make sure your calculator is in radian mode. θ 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.50 1.65 2 2.4 tanθ θ 2.8 3.2 3.6 4.0 4.4 4.6 4.65 4.78 5.0 5.6 6.0 6.28 tanθ 2. Use the table above to graph tanθ.Mathematics Learning Centre, University ofSydney 12 Your graph should look like this for values of θ between 0 and π. Notice that there is a vertical asymptote π at θ = . This is because tan θ is not 2 π defined at θ = .You will find another 2 3π vertical asymptote at θ = . When θ = 2 Oππ /2 0or π, tanθ =0. For θ greater than 0 π and less than , tanθ is positive. For 2 π values of θ greater than and less than 2 π, tanθ is negative. 4.4 Extending the domain The definitions of sine, cosine and tangent can be extended to all real values of θ in the following way. 5π π 1 =2π + corresponds to the arc length of 1 revolu- 4 2 2 B tions around the unit circle going anticlockwise from A to B. Since B has coordinates (0, 1) we can use the previous definitions to get: A O 5π sin =1, 2 5π cos =0, 2 5π tan is undefined. 2 Similarly, −16 ≈−2.546× 2π B = −2× 2π +−0.546× 2π, sin(−16) ≈ sin(−0.546× 2π) A O ≈ 0.29, cos(−16) ≈−0.96, tan(−16) ≈−0.30.Mathematics Learning Centre, University ofSydney 13 4.4.1 Exercise Evaluate the following trig functions giving exact answers where you are able. 15π 13π −14π 23π 1. sin 2. tan 3. cos 15 4. tan 5. sin 2 6 3 6 Notice The values of sine and cosine functions repeat after every interval of length 2π. Since the real numbers x, x+2π, x− 2π, x+4π, x− 4π etc differ by a multiple of 2π, they correspond to the same point on the unit circle. So, sinx = sin(x+2π)= sin(x− 2π)= sin(x+4π)= sin(x− 4π) etc. We can see the effect of this in the functions below and will discuss it further in the next chapter. sinθ 1 θ −2π −π 0π 2π −1 cosθ 1 θ −2π −π 0π 2π −1 The tangent function repeats after every interval of length π. tanθ 1 θ −2π −π 0π 2π −1Mathematics Learning Centre, University ofSydney 14 5 Graphs of Trigonometric Functions In this section we use our knowledge of the graphs y = sin x and y = cos x to sketch the graphs of more complex trigonometric functions. sin x 1 x −2π −π 0π 2π −1 cosx 1 x −2π −π 0π 2π −1 Let’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 unit circle. 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 called periodic functions. Functions with the property that f(x)= f(x + a) for all x are called periodic functions. 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 functions that are not trigonometric functions. For example, the function below is periodic with period 2. 2 2 1 0 -4 -3 -2 -1 1234Mathematics Learning Centre, University ofSydney 15 The values of the functions y = sin x and y = cos x oscillate between−1 and 1. We say that y = sin x and y = cos x have amplitude 1. A general definition for the amplitude of any periodic function is: The amplitude of a periodic function is half the distance between its minimum and maximum values. Also, the functions y = sin x and y = cos x oscillate about the x-axis. We refer to the x-axis as the mean level of these functions, or say that they have a mean level of 0. We notice that the graphs of sinx and cosx have the same shape. The graph of sinx looks π like the graph of cosx shifted to the right by units. We say that the phase difference 2 π between the two functions is . 2 Other trigonometric functions can be obtained by modifying the graphs of sinx and cosx. 5.1 Changing the amplitude Consider the graph of the function y =2 sin x. y y = 2sinx 1 x y = sinx −2π −π 0π 2π −1 The graph of y =2 sin x has the same period as y = sin x but has been stretched in the y direction by a factor of 2. That is, for every value of x the y value for y =2 sin x is twice the y value for y = sin x. So, the amplitude of the function y =2 sin x is 2. Its period is 2π. In general we can say that the amplitude of the function y = a sin x is a, since in this case y = a sin x oscillates between−a and a. What happens if a is negative? See the solution to number 3 of the following exercise. 5.1.1 Exercise Sketch the graphs of the following functions. 1. y =3 cos x 1 2. y = sin x 2 3. y =−3 cos xMathematics Learning Centre, University ofSydney 16 5.2 Changing the period Let’s consider the graph of y = cos 2x. To sketch the graph of y = cos 2x, first think about some specific points. We will look at the points where the function y = cos x equals 0 or±1. y y = cos 2x y = cos x 1 x −2π −π 0π 2π −1 cos x=1 when x=0, so cos 2x=1 when 2x=0, ie x=0. π π π cos x=0 when x =,so cos 2x=0 when 2x =,ie x = . 2 2 4 π cos x =−1 when x = π,so cos 2x =−1 when 2x = π,ie x = . 2 3π 3π 3π cos x=0 when x =,so cos 2x=0 when 2x = ,ie x = . 2 2 4 cos x=1 when x=2π,so cos 2x=1 when 2x=2π,ie x = π. As we see from the graph, the function y = cos 2x has a period of π. The function still oscillates between the values−1 and 1, so its amplitude is 1. 1 Now, let’s consider the function y = cos x. Again we will sketch the graph by looking at 2 the points where y = cos x equals 0 or±1. y y = cos 1/2x y = cos x 1 x −π 0π 2π 3π 4π −1 1 1 cos x=1 when x=0, so cos x=1 when x=0, ie x=0. 2 2 π 1 1 π cos x=0 when x =,so cos x=0 when x =,ie x = π. 2 2 2 2 1 1 cos x =−1 when x = π,so cos x =−1 when x = π,ie x=2π. 2 2 3π 1 1 3π cos x=0 when x =,so cos x=0 when x = ,ie x=3π. 2 2 2 2 1 1 cos x=1 when x=2π,so cos x=1 when x=2π,ie x=4π. 2 2Mathematics Learning Centre, University ofSydney 17 1 In this case our modified function y = cos x has period 4π. It’s amplitude is 1. 2 What happens if we take the function y = cosωx where ω 0? y y = cos xω 1 x −π/ω 0 −2π/ωπ/ω 2π/ω −1 cos x=1 when x=0, so cosωx=1 when ωx=0, ie x=0. π π π cos x=0 when x =,so cosωx=0 when ωx =,ie x = . 2 2 2ω π cos x =−1 when x = π,so cosωx =−1 when ωx = π,ie x = . ω 3π 3π 3π cos x=0 when x =,so cosωx=0 when ωx = ,ie x = . 2 2 2ω 2π cos x=1 when x=2π,so cosωx=1 when ωx=2π,ie x = . ω In general, if we take the function y = cosωx where ω 0, the period of the function is 2π . ω What happens if we have a function like y = cos(−2x)? See the solution to number 3 of the following exercise. 5.2.1 Exercise Sketch the graphs of the following functions. Give the amplitude and period of each function. 1 1. y = cos x 2 x 2. y =2 sin 4 3. y = cos(−2x) 4. y = sin(−2x) 5. y =3 sinπx 1 6. y =− sin 2πx 2 7. Find the equation of a sin or cos function which has amplitude 4 and period 2.Mathematics Learning Centre, University ofSydney 18 5.3 Changing the mean level We saw above that the functions y = sin x and y = cos x both oscillate about the x-axis which is sometimes refered to as the mean level for y = sin x and y = cos x. We can change the mean level of the function by adding or subtracting a constant. For example, adding the constant 2 to y = cos x gives us y = cos x+2 and has the effect of shifting the whole graph up by 2 units. So, the mean level of y = cos x+2 is 2. y 2 1 2 x −2π −π 0π 2π Similarly, we can shift the graph of y = cos x down by two units. In this case, we have y = cos x− 2, and this function has mean level−2. y x −2π −π 0π 2π 2 -2 y = cos x – 2 In general, if d 0, the function y = cos x + d looks like the function y = cos x shifted up by d units. If d 0, then the function y = cos x− d looks like the function y = cos x shifted down by d units. If d 0, say d =−2, the function y = cos x + d = cos x+(−2) can be writen as y = cos x−2so again looks like the function y = cos x shifted down by 2 units. 5.3.1 Exercise Sketch the graphs of the following functions. 1. y = sin 2x+3 2. y =2 cosπx− 1 3. Find a cos or sin function which has amplitude 2, period 1, and mean level−1.Mathematics Learning Centre, University ofSydney 19 5.4 Changing the phase π Consider the function y = sin(x− ). 4 y y = sin (x - )π/4 y = sin x x −2π −π 0π 2π π/4 π To sketch the graph of y = sin(x− )we again use the points at which y = sin x is 0 or 4 ±1. π π π π sin x=0 when x=0, so sin(x− )=0 when x− =0, ie x=0+ = . 4 4 4 4 π π π π π π 3π sin x=1 when x =,so sin(x− )=1 when x− = ,ie x = + = . 2 4 4 2 2 4 4 π π π 5π sin x=0 when x = π,so sin(x− )=0 when x− = π,ie x = π + = . 4 4 4 4 3π π π 3π 3π π 7π sin x =−1 when x = ,so sin(x− )=−1 when x− = ,ie x = + = . 2 4 4 2 2 4 4 π π π 9π sin x=0 when x=2π,so sin(x− )=0 when x− =2π,ie x=2π + = . 4 4 4 4 π π The graph of y = sin(x− )looks like the graph of y = sin x shifted units to the right. 4 4 π We say that there has been a phase shift to the right by . 4 π Now consider the function y = sin(x + ). 4 y y = sin (x + )π/4 π/4 y = sin x x −2π −π 0π 2π π π π π sin x=0 when x=0, so sin(x + )=0 when x + =0, ie x=0− =− . 4 4 4 4 π π π π π π π sin x=1 when x =,so sin(x + )=1 when x + = ,ie x = − = . 2 4 4 2 2 4 4 π π π 3π sin x=0 when x = π,so sin(x + )=0 when x + = π,ie x = π− = . 4 4 4 4 3π π π 3π 3π π 5π sin x =−1 when x = ,so sin(x + )=−1 when x + = ,ie x = − = . 2 4 4 2 2 4 4 π π π 7π sin x=0 when x=2π,so sin(x + )=0 when x + =2π,ie x=2π− = . 4 4 4 4

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