Inverse Trigonometric functions problems

inverse trigonometric functions and their derivatives and inverse trigonometric functions answers and inverse trigonometric functions integration
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8 Periodic Functions Figure 1 (credit: "maxxer_", Flickr) ChAPTeR OUTl Ine 8.1 graphs of the Sine and Cosine Functions 8.2 graphs of the Other Trigonometric Functions 8.3 Inverse Trigonometric Functions Introduction Each day, the sun rises in an easterly direction, approaches some maximum height relative to the celestial equator, and sets in a westerly direction. The celestial equator is an imaginary line that divides the visible universe into two halves in much the same way Earth’s equator is an imaginary line that divides the planet into two halves. The exact path the sun appears to follow depends on the exact location on Earth, but each location observes a predictable pattern over time. e pa Th ttern of the sun’s motion throughout the course of a year is a periodic function. Creating a visual representation of a periodic function in the form of a graph can help us analyze the properties of the function. In this chapter, we will investigate graphs of sine, cosine, and other trigonometric functions. 641642 CHAPTER 8 Perio dic uf Nctio Ns l eARnIng Obje CTIveS In this section, you will: • Graph variations of y = sin(x ) and y = cos(x ). • Use phase shifts of sine and cosine curves. 8.1 gRAPhS OF The SIne And C OSIne F UnCTIOnS Figure 1 l ight can be separated into colors because of its wavelike properties. (credit: "wonderferret"/ Flickr) White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can be seen only when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow. Light waves can be represented graphically by the sine function. In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions. graphing Sine and Cosine Functions Recall that the sine and cosine functions relate real number values to the x- and y-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let’s start with the sine function. We can create a table of values and use them to sketch a graph. Table 1 lists some of the values for the sine function on a unit circle. 5π π π π π 2π 3π _ _ _ _ _ _ _         x 0 π 6 4 3 2 3 4 6 — — — — 1 1 √ 2 √ 3 √3 √2 _ _ _ _ _ _ sin(x) 0 1 0 2 2 2 2 2 2 Table 1 Plotting the points from the table and continuing along the x-axis gives the shape of the sine function. See Figure 2. y 2 y = sin (x) 1 x π π 3π 5π 3π 7π π 2π 4 2 4 4 2 4 –1 –2 Figure 2 The sine function8.1 g ra Phs of t he s i Ne a Nd c osi Ne f u Ncti o Ns 643 Notice how the sine values are positive between 0 and π, which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between π and 2π, which correspond to the values of the sine function in quadrants III and IV on the unit circle. See Figure 3. y y = sin (x) 1 0 x π π π π 6 4 3 2 –1 Figure 3 Plotting values of the sine function Now let’s take a similar look at the cosine function. Again, we can create a table of values and use them to sketch a graph. Table 2 lists some of the values for the cosine function on a unit circle. π π π π 2π 3π 5π _ _ _ _ _ _ _         x 0 π 4 3 2 3 4 6 6 — — — — 1 1 √3 √ 2 _ _ √ 2 √ 3 _ _ _ _ − cos(x) 1 0 − − −1 2 2 2 2 2 2 Table 2 As with the sine function, we can plots points to create a graph of the cosine function as in Figure 4. y 1 y = cos (x) 2π 3π 5π 3 4 6 0 x π π π π 5π 3π 7π 2π π 6 4 3 2 4 2 4 –1 Figure 4 The cosine function Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval −1, 1. In both graphs, the shape of the graph repeats after 2 π, which means the functions are periodic with a period of 2π. A periodic function is a function for which a specic h fi orizontal shift, P, results in a function equal to the original function: f (x + P) = f (x) for all values of x in the domain of f . When this occurs, we call the smallest such horizontal shift with P 0 the period of the function. Figure 5 shows several periods of the sine and cosine functions. y y 1 period 1 period y = sin (x) y = cos (x) 1 1 –3π x x –2π –π π 2π –3π –2π –π π 2π 3π 3π –1 –1 Figure 5644 CHAPTER 8 Perio dic fu Nctio Ns Looking again at the sine and cosine functions on a domain centered at the y-axis helps reveal symmetries. As we can see in Figure 6, the sine function is symmetric about the origin. Recall from The Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because sin(−x) = −sin x. Now we can clearly see this property from the graph. y 2 y = sin (x) 1 x –2π –π π 2π –1 –2 Figure 6 Odd symmetry of the sine function Figure 7 shows that the cosine function is symmetric about the y-axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that cos(−x) = cos x. y 2 1 x –2π –ππ 2π –1 y = cos (x) –2 Figure 7 even symmetry of the cosine function characteristics of sine and cosine functions e s Th i ne and cosine functions have several distinct characteristics: • e Th y are periodic functions with a period of 2 π. • e do Th main of each function is ( −∞, ∞) and the range is −1, 1. • e g Th raph of y = sin x is symmetric about the origin, because it is an odd function. • e g Th raph of y = cos x is symmetric about the y- axis, because it is an even function. Investigating Sinusoidal Functions As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or cosine function is known as a sinusoidal function. The general forms of sinusoidal functions are y = Asin(Bx − C) + D and y = Acos(Bx − C) + D8.1 g ra Phs of t he s i Ne a Nd c osi Ne f u Ncti o Ns 645 Determining the Period of Sinusoidal Functions Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period. 2π _ In the general formula, B is related to the period by P = . I f B 1, then the period is less than 2π and the function ∣ ∣ B ∣ ∣ undergoes a horizontal compression, whereas if B 1, then the period is greater than 2π and the function undergoes ∣ ∣ a horizontal stretch. For example, f (x) = sin(x), B = 1, so the period is 2π, which we knew. If f (x) = sin(2x), then B = 2, x 1 _ _ so the period is π and the graph is compressed. If f (x) = sin , then B = , so the period is 4π and the graph is   2 2 stretched. Notice in Figure 8 how the period is indirectly related to B . ∣ ∣ y f (x) = sin (x) 1 x f (x) = sin( ) 2 x π 3π π 2π 2 2 –1 f (x) = sin (2x) Figure 8 period of sinusoidal functions If we let C = 0 and D = 0 in the general form equations of the sine and cosine functions, we obtain the forms y = Asin(Bx) y = Acos(Bx) 2π _ e p Th eriod is . B ∣ ∣ Example 1 Identifying the Period of a Sine or Cosine Function π _ Determine the period of the function f (x) = sin x .   6 Solution Let’s begin by comparing the equation to the general form y = Asin(Bx). π _ In the given equation, B = , so the period will be 6 2π _ P = B ∣ ∣ 2π _ = π _ 6 6 _ = 2π ⋅ π = 12 Try It 1 x __ Determine the period of the function g(x) = cos .   3 Determining Amplitude Returning to the general formula for a sinusoidal function, we have analyzed how the variable B relates to the period. Now let’s turn to the variable A so we can analyze how it is related to the amplitude, or greatest distance from rest. A represents the vertical stretch factor, and its absolute value A i s the amplitude. The local maxima will be a distance A above the ∣ ∣ ∣ ∣ vertical midline of the graph, which is the line x = D ; because D = 0 in this case, the midline is the x-axis. The local minima will be the same distance below the midline. If A 1, the function is stretched. For example, the amplitude of ∣ ∣ f (x) = 4sin x is twice the amplitude of f (x) = 2sin x. If A 1, the function is compressed. Figure 9 compares several ∣ ∣ sine functions with die ff rent amplitudes.646 CHAPTER 8 Perio dic fu Nctio Ns y f (x) = 4 sin(x) 4 f (x) = 3 sin(x) 3 f (x) = 2 sin(x) f (x) = 1 sin(x) 2 9π 5π 1 3π 7π 11π – – 2 2 2 2 2 11π 7π 3π 5π 9π – – – –1 2 2 2 2 2 –2 –3 –4 Figure 9 amplitude of sinusoidal functions If we let C = 0 and D = 0 in the general form equations of the sine and cosine functions, we obtain the forms y = Asin(Bx) and y = Acos(Bx) The amplitude is A, and the vertical height from the midline is A . I n addition, notice in the example that ∣ ∣ 1 __ A = amplitude = maximum − minimum ∣ ∣ ∣ ∣ 2 Example 2 Identifying the Amplitude of a Sine or Cosine Function What is the amplitude of the sinusoidal function f (x) = −4sin(x)? Is the function stretched or compressed vertically? Solution Let’s begin by comparing the function to the simplie fi d form y = Asin(Bx). In the given function, A = −4, so the amplitude is A = −4 = 4. The function is stretched. ∣ ∣ ∣ ∣ Analysis The negative value of A results in a reflection across the x-axis of the sine function, as shown in Figure 10. y f (x) = −4 sin x 4 3 2 x 3π π π 3π – – 2 2 2 2 –1 –2 –3 –4 Figure 10 Try It 2 1 __ What is the amplitude of the sinusoidal function f (x) = sin(x)? Is the function stretched or compressed vertically ? 2 Analyzing graphs of variations of y = sin x and y = cos x Now that we understand how A and B relate to the general form equation for the sine and cosine functions, we will explore the variables C and D. Recall the general form: y = Asin(Bx − C) + D and y = Acos(Bx − C) + D or C C __ __ y = Asin B x − + D and y = Acos B x − + D       B B8.1 g ra Phs of t he s i Ne a Nd c osi Ne f u Ncti o Ns 647 C __ The value for a sinusoidal function is called the phase shift , or the horizontal displacement of the basic sine or B cosine function. If C 0, the graph shifts to the right. If C 0, the graph shifts to the left. The greater the value of C ∣ ∣ , the more the graph is shifted. Figure 11 shows that the graph of f (x) = sin(x − π) shifts to the right by π units, which π π __ __ is more than we see in the graph of f (x) = sin x − , which shifts to the right by units.   4 4 y π f (x) = sin x − 4 f (x) = sin(x) 1 f (x) = sin(x − π) x π 3π 5π π 2π 3π 2 2 2 Figure 11 While C relates to the horizontal shift, D indicates the vertical shift from the midline in the general formula for a sinusoidal function. See Figure 12. The function y = cos(x) + D has its midline at y = D. y y = A sin(x) + D Midline y = D x π 2π 3π Figure 12 Any value of D other than zero shifts the graph up or down. Figure 13 compares f (x) = sin x with f (x) = sin x + 2, which is shifted 2 units up on a graph. y f (x) = sin(x) + 2 3 2 f (x) = sin(x) 1 x π 3π 5π π 2π 3π 2 2 2 –1 Figure 13 variations of sine and cosine functions C __ Given an equation in the form f (x) = Asin(Bx − C) + D or f (x) = Acos(Bx − C) + D, is the phase shift and B D is the vertical shift. Example 3 Identifying the Phase Shift of a Function π __ Determine the direction and magnitude of the phase shift for f (x) = sin x + − 2.   6 Solution Let’s begin by comparing the equation to the general form y = Asin(Bx − C) + D.648 CHAPTER 8 Perio dic uf Nctio Ns π __ In the given equation, notice that B = 1 and C = − . So the phase shift is 6 π __ C 6 __ _ = − B 1 π __ = − 6 π __ or units to the left. 6 Analysis We must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation π π __ __ shows a minus sign before C. Therefore f (x) = sin x +   − 2 can be rewritten as f (x) = sin x − −   − 2.      6 6 If the value of C is negative, the shift is to the left. Try It 3 π _ Determine the direction and magnitude of the phase shift for f (x) = 3cos x − .   2 Example 4 Identifying the Vertical Shift of a Function Determine the direction and magnitude of the vertical shift for f (x) = cos(x) − 3. Solution Let’s begin by comparing the equation to the general form y = Acos(Bx − C) + D. In the given equation, D = −3 so the shift is 3 units downward. Try It 4 Determine the direction and magnitude of the vertical shift for f (x) = 3sin(x) + 2. How To… Given a sinusoidal function in the form f (x) = Asin(Bx − C) + D, identify the midline, amplitude, period, and phase shift. 1. Determine the amplitude as ∣ A ∣. 2π _ 2. Determine the period as P = . B ∣ ∣ C __ 3. Determine the phase shift as . B 4. Determine the midline as y = D. Example 5 Identifying the V ariations of a Sinusoidal Function from an Equation Determine the midline, amplitude, period, and phase shift of the function y = 3sin(2x) + 1. Solution Let’s begin by comparing the equation to the general form y = Asin(Bx − C) + D. A = 3, so the amplitude is A = 3. ∣ ∣ 2π 2π _ __ Next, B = 2, so the period is P = = = π. 2 ∣ B ∣ C 0 __ __ Th ere is no added constant inside the parentheses, so C = 0 and the phase shift is = = 0. B 2 Finally, D = 1, so the midline is y = 1. Analysis Inspecting the graph, we can determine that the period is π, the midline is y = 1, and the amplitude is 3. See Figure 14.8.1 g ra Phs of t he s i Ne a Nd c osi Ne f u Ncti o Ns 649 y Amplitude: A = 3 4 3 y = 3 sin (2x) + 1 2 Midline: y = 1 x π 3π π 2π 2 2 –1 –2 Period = π Figure 14 Try It 5 1 x π __ __ __ Determine the midline, amplitude, period, and phase shift of the function y = cos –   .   2 3 3 Example 6 Identifying the Equation for a Sinusoidal Function from a Graph Determine the formula for the cosine function in Figure 15. y 1 0.5 x –2π –π π 2π 3π 4π Figure 15 Solution To determine the equation, we need to identify each value in the general form of a sinusoidal function. y = Asin(Bx − C) + D y = Acos(Bx − C) + D e g Th raph could represent either a sine or a cosine function that is shifted and/or ree fl cted. When x = 0, the graph has an extreme point, (0, 0). Since the cosine function has an extreme point for x = 0, let us write our equation in terms of a cosine function. Let’s start with the midline. We can see that the graph rises and falls an equal distance above and below y = 0.5. This value, which is the midline, is D in the equation, so D = 0.5. e g Th r eatest distance above and below the midline is the amplitude. The maxima are 0.5 units above the midline and the minima are 0.5 units below the midline. So A = 0.5. Another way we could have determined the amplitude is ∣ ∣ 1 __ by recognizing that the die ff rence between the height of local maxima and minima is 1, so A = = 0.5. Also, the ∣ ∣ 2 graph is ree fl cted about the x-axis so that A = −0.5. e g Th raph is not horizontally stretched or compressed, so B = 1; and the graph is not shifted horizontally, so C = 0. Putting this all together, g(x) = −0.5cos(x) + 0.5 y Try It 6 3 Determine the formula for the sine function in Figure 16. 2 1 x –2π –π π 2π Figure 16 650 CHAPTER 8 Perio dic uf Nctio Ns Example 7 Identifying the Equation for a Sinusoidal Function from a Graph Determine the equation for the sinusoidal function in Figure 17. y 1 x –5 –3 –11 357 –1 –2 –3 –4 –5 Figure 17 Solution With the highest value at 1 and the lowest value at −5, the midline will be halfway between at −2. So D = −2. e d Th i stance from the midline to the highest or lowest value gives an amplitude of A = 3. ∣ ∣ The period of the graph is 6, which can be measured from the peak at x = 1 to the next peak at x = 7, or from the 2π _ distance between the lowest points. Therefore, P = = 6. Using the positive value for B, we find that B ∣ ∣ 2π 2π π __ __ __ B = = = P 6 3 π π __ __ So far, our equation is either y = 3sin x − C − 2 or y = 3cos x − C − 2. For the shape and shift, we have more     3 3 than one option. We could write this as any one of the following: • a cosine shifted to the right • a negative cosine shifted to the left • a sine shifted to the left • a negative sine shifted to the right While any of these would be correct, the cosine shifts are easier to work with than the sine shifts in this case because they involve integer values. So our function becomes π π π 2π __ __ __ __ y = 3cos x − − 2 or y = −3cos x + −2     3 3 3 3 Again, these functions are equivalent, so both yield the same graph. Try It 7 Write a formula for the function graphed in Figure 18. y 8 6 4 2 x –9 –7 –5 –3 –1 1357 911 Figure 188.1 g ra Phs of t he s i Ne a Nd c osi Ne f u Ncti o Ns 651 graphing variations of y = sin x and y = cos x Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create graphs from equations. Instead of focusing on the general form equations y = Asin(Bx − C) + D and y = Acos(Bx − C) + D, we will let C = 0 and D = 0 and work with a simplie fi d form of the equations in the following examples. How To… Given the function y = Asin(Bx), sketch its graph. 1. Identify the amplitude, A . ∣ ∣ 2π ___ 2. Identify the period, P = . B ∣ ∣ 3. Start at the origin, with the function increasing to the right if A is positive or decreasing if A is negative. π ____ 4. At x = there is a local maximum for A 0 or a minimum for A 0, with y = A. 2 B ∣ ∣ π ___ 5. e c Th urve returns to the x-axis at x = . B ∣ ∣ 3π ____ 6. e Th re is a local minimum for A 0 (maximum for A 0 ) at x = with y = −A. 2 B ∣ ∣ π ____ 7. e c Th urve returns again to the x-axis at x = . 2 B ∣ ∣ Example 8 Graphing a Function and Identifying the Amplitude and Period πx __ Sketch a graph of f (x) = −2sin .   2 Solution Let’s begin by comparing the equation to the form y = Asin(Bx). Step 1. We can see from the equation that A = −2, so the amplitude is 2. A = 2 ∣ ∣ π __ Step 2. The equation shows that B = , so the period is 2 2π _ P = π __ 2 2 __ = 2π ⋅ π = 4 Step 3. Because A is negative, the graph descends as we move to the right of the origin. Step 4–7. The x-intercepts are at the beginning of one period, x = 0, the horizontal midpoints are at x = 2 and at the end of one period at x = 4. e q Th uarter points include the minimum at x = 1 and the maximum at x = 3. A local minimum will occur 2 units below the midline, at x = 1, and a local maximum will occur at 2 units above the midline, at x = 3. Figure 19 shows the graph of the function. y πx y = f (x) = −2sin 2 2 1 x –2 –1 123456 –1 –2 Figure 19652 CHAPTER 8 Perio dic fu Nctio Ns Try It 8 Sketch a graph of g(x) = −0.8cos(2x). Determine the midline, amplitude, period, and phase shift. How To… Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph. 1. Express the function in the general form y = Asin(Bx − C) + D or y = Acos(Bx − C) + D. 2. Identify the amplitude, A . ∣ ∣ 2π ___ 3. Identify the period, P = . B ∣ ∣ C __ 4. Identify the phase shift, . B C __ 5. Draw the graph of f (x) = Asin(Bx) shifted to the right or left by and up or down by D. B Example 9 Graphing a Transformed Sinusoid π π __ __ Sketch a graph of f (x) = 3sin x − .   4 4 Solution π π __ __ Step 1. The function is already written in general form: f (x) = 3sin x − . Th is graph will have the shape of a sine   4 4 function, starting at the midline and increasing to the right. Step 2. A = 3 = 3. The amplitude is 3. ∣ ∣ ∣ ∣ π π __ __ Step 3. Since B = = , we determine the period as follows. ∣ ∣   4 4 2π 2π 4 ___ _ _ P = = = 2π ⋅ = 8 π π B __ ∣ ∣ 4 e p Th eriod is 8. π __ Step 4. Since C = , the phase shift is 4 π _ C 4 __ _ = = 1. π B _ 4 e p Th hase shift is 1 unit. Step 5. Figure 20 shows the graph of the function. y π π x − f (x) = 3 sin 3 4 4 2 1 x –7 –5 –3 –11 3579 –1 –2 –3 Figure 20 A horizontally compressed, vertically stretched, and horizontally shifted sinusoid8.1 g ra Phs of t he s i Ne a Nd c osi Ne f u Ncti o Ns 653 Try It 9 π π __ __ Draw a graph of g(x) = −2cos x + . Determine the midline, amplitude, period, and phase shift.   3 6 Example 10 Identifying the Properties of a Sinusoidal Function π __ Given y = −2cos x + π + 3, determine the amplitude, period, phase shift, and horizontal shift. Then graph the   2 function. Solution Begin by comparing the equation to the general form and use the steps outlined in Example 9. y = Acos(Bx − C) + D Step 1. The function is already written in general form. Step 2. Since A = −2, the amplitude is ∣ A ∣ = 2. π 2π 2π 2 __ ___ _ __ Step 3. B = , so the period is P = = = 2π ⋅ = 4. The period is 4. ∣ ∣ π 2 B __ π ∣ ∣ 2 C −π 2 __ _ __ Step 4. C = −π, so we calculate the phase shift as = = −π ⋅ = −2. The phase shift is −2. π B __ π 2 Step 5. D = 3, so the midline is y = 3, and the vertical shift is up 3. Since A is negative, the graph of the cosine function has been ree fl cted about the x-axis. Figure 21 shows one cycle of the graph of the function. y π y = −2 cos x + π + 3 2 5 Amplitude = 2 4 3 Midline: y = 3 2 1 Period = 4 x –6 –4 –22 –1 Figure 21 Using Transformations of Sine and Cosine Functions We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, circular motion can be modeled using either the sine or cosine function. Example 11 Finding the Vertical Component of Circular Motion A point rotates around a circle of radius 3 centered at the origin. Sketch a graph of the y-coordinate of the point as a function of the angle of rotation. Solution Recall that, for a point on a circle of radius r, the y-coordinate of the point is y = r sin(x), so in this case, we get the equation y(x) = 3 sin(x). The constant 3 causes a vertical stretch of the y-values of the function by a factor of 3, which we can see in the graph in Figure 22. y y(x) = 3 sin x 3 2 1 x π 3π 5π 7π 2 2 2 2 –1 –2 –3 Figure 22654 CHAPTER 8 Perio dic fu Nctio Ns Analysis Notice that the period of the function is still 2π ; as we travel around the circle, we return to the point (3, 0) for x = 2π, 4π, 6π, ... Because the outputs of the graph will now oscillate between –3 and 3, the amplitude of the sine wave is 3. Try It 10 What is the amplitude of the function f (x) = 7cos(x)? Sketch a graph of this function. Example 12 Finding the Vertical Component of Circular Motion A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled P, as shown in Figure 23. Sketch a graph of the height above the ground of the point P as the circle is rotated; then find a function that gives the height in terms of the angle of rotation. 3  4  P Figure 23 Solution Sketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft, as shown in Figure 24. y y = −3 cos x + 4 7 6 5 Midline: y = 4 3 2 x π 2π 3π 4π Figure 24 Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other. Let’s use a cosine function because it starts at the highest or lowest value, while a sine function starts at the middle value. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical ree fl ction. Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example. Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4. Putting these transformations together, we find that y = −3cos(x) + 48.1 g ra Phs of t he s i Ne a Nd c osi Ne f u Ncti o Ns 655 Try It 11 A weight is attached to a spring that is then hung from a board, as shown in Figure 25. As the spring oscillates up and down, the position y of the weight relative to the board ranges from −1 in. (at time x = 0) to −7 in. (at time x = π) below the board. Assume the position of y is given as a sinusoidal function of x. Sketch a graph of the function, and then find a cosine function that gives the position y in terms of x. y Figure 25 Example 13 Determining a Rider’s Height on a Ferris Wheel The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider’s height above ground as a function of time in minutes. Solution With a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below the center. Passengers board 2 m above ground level, so the center of the wheel must be located 67.5 + 2 = 69.5 m above ground level. The midline of the oscillation will be at 69.5 m. e w Th heel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes. Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically ree fl cted cosine curve. • Amplitude: 67.5, so A = 67.5 • Midline: 69.5, so D = 69.5 2π π ___ __ • Period: 30, so B = = 30 15 • Shape: −cos(t) An equation for the rider’s height would be π __ y = − 67.5cos t + 69.5   15 where t is in minutes and y is measured in meters. Access these online resources for additional instruction and practice with graphs of sine and cosine functions. • Amplitude and Period of Sine and Cosine (http://openstaxcollege.org/l/ampperiod) • Translations of Sine and Cosine (http://openstaxcollege.org/l/translasincos) • graphing Sine and Cosine Transformations (http://openstaxcollege.org/l/transformsincos) • graphing the Sine Function (http://openstaxcollege.org/l/graphsinefunc)656 CHAPTER 8 Perio dic uf Nctio Ns 8.1 SeCTIOn exe RCISeS veRbAl 1. Why are the sine and cosine functions called 2. How does the graph of y = sin x compare with periodic functions? the graph of y = cos x? Explain how you could horizontally translate the graph of y = sin x to obtain y = cos x. 3. For the equation Acos(Bx + C) + D, what constants 4. How does the range of a translated sine function ae ff ct the range of the function and how do they relate to the equation y = Asin(Bx + C) + D? ae ff ct the range? 5. How can the unit circle be used to construct the graph of f(t) = sin t? gRAPhICAl For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for x 0. Round answers to two decimal places if necessary. 2 __ 6. f (x) = 2sin x 7. f (x) =   cos x 8. f (x) = −3sin x 3 9. f (x) = 4sin x 10. f (x) = 2cos x 11. f (x) = cos(2x) 1 6 __ __ 13. f (x) = 4cos(πx) 12. f (x) = 2sin x 14. f (x) = 3cos x     2 5 15. y = 3sin(8(x + 4)) + 5 16. y = 2sin(3x − 21) + 4 17. y = 5sin(5x + 20) − 2 For the following exercises, graph one full period of each function, starting at x = 0. For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for x  0. State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary. 5π π π ___ __ __ 18. f(t) = 2sin t − 19. f(t) = −cos t + + 1 20. f (t) = 4cos 2 t + − 3        6 3 4 1 5π π __ ___ _ 21. f(t) = −sin t + 22. f (x) = 4sin (x − 3) + 7     2 3 2 23. Determine the amplitude, midline, period, and an 24. Determine the amplitude, period, midline, and an equation involving the sine function for the graph equation involving cosine for the graph shown in shown in Figure 26. Figure 27. f(x) f(x) 5 5 4 4 3 3 2 2 1 1 x x 3π π π 3π –5 –4–3 –2 –1 14 2 3 5 –π π – – –1 –1 2 2 2 2 –2 –2 –3 –3 –4 –4 –5 –5 Figure 26 Figure 27 8.1 s ectio N e xercises 657 25. Determine the amplitude, period, midline, and an 26. Determine the amplitude, period, midline, and equation involving cosine for the graph shown in an equation involving sine for the graph shown in Figure 28. Figure 29. f(x) f(x) 5 5 4 4 3 3 2 2 1 1 x x 20 16 12 8 4 41 8 12 6 20 –6–5 –4–3 –2 –1 14 2 3 5 6 – – – – – –1 –1 –2 –2 –3 –3 –4 –4 –5 –5 Figure 28 Figure 29 27. Determine the amplitude, period, midline, and an 28. Determine the amplitude, period, midline, and equation involving cosine for the graph shown in an equation involving sine for the graph shown in Figure 30. Figure 31. f(x) f(x) 5 4 4 3 2 2 1 x x π π −2 + π –5 –4–3 –2 –1 14 2 3 5 − 2 –1 2 2 –2 –2 –3 –4 –4 –5 Figure 30 Figure 31 29. Determine the amplitude, period, midline, and an 30. Determine the amplitude, period, midline, and equation involving cosine for the graph shown in an equation involving sine for the graph shown in Figure 32. Figure 33. f(x) f(x) 5 5 4 4 3 3 2 2 1 1 x x 5 4 3 2 1 14 2 3 5 5 4 3 2 1 14 2 3 5 – – – – – – – – – – –1 –1 –2 –2 –3 –3 –4 –4 –5 –5 Figure 32 Figure 33 658 CHAPTER 8 Perio dic uf Nctio Ns Algeb RAIC For the following exercises, let f (x) = sin x. 1 __ 31. On 0, 2π), solve f (x) = 0. 32. On 0, 2π), solve f (x) =    . 2 — π √2 __ ____ 33. Evaluate f . 34. On 0, 2π), f (x) =    . Fin d all values of x.   2 2 35. On 0, 2π), the maximum value(s) of the function 36. On 0, 2π), the minimum value(s) of the function occur(s) at what x-value(s)? occur(s) at what x-value(s)? 37. Show that f(−x) = −f (x). This m eans that f (x) = sin x is an odd function and possesses symmetry with respect to ________________. For the following exercises, let f (x) = cos x. 1 __ 38. On 0, 2π), solve the equation f (x) = cos x = 0. 39. On 0, 2π), solve f (x) =   . 2 40. On 0, 2π), find the x-intercepts of f (x) = cos x. 41. On 0, 2π), find the x-values at which the function has a maximum or minimum value. — √ 3 ____ 42. On 0, 2π), solve the equation f (x) = . 2 TeChn Ol Ogy 43. Graph h(x) = x + sin x on 0, 2π. Explain why the 44. Graph h(x) = x + sin x on −100, 100. Did the graph appears as it does. graph appear as predicted in the previous exercise? 45. Graph f (x) = x sin x on 0, 2π and verbalize how 46. Graph f (x) = x sin x on the window −10, 10 and the graph varies from the graph of f (x) = sin x. explain what the graph shows. sin x ____ 47. Graph f (x) = on the window −5π, 5π and x explain what the graph shows. ReAl-W ORld A PPl ICATIOnS 48. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function h(t) gives a person’s height in meters above the ground t minutes aer t ft he wheel begins to turn. a. Find the amplitude, midline, and period of h(t). b. Find a formula for the height function h(t). c. How high off the ground is a person after 5 minutes?8.2 g ra Phs of t he o the r t rig o Nome tric f u Ncti o Ns 659 l eARnIng Obje CTIveS In this section, you will: • Analyze the graph of y = tan x. • Graph variations of y = tan x. • Analyze the graphs of y = sec x and y = csc x. • Graph variations of y = sec x and y = csc x. • Analyze the graph of y = cot x. • Graph variations of y = cot x. 8.2 gRAPhS OF The OTheR TRIgOnOmeTRIC FUnCTIOnS We know the tangent function can be used to find distances, such as the height of a building, mountain, or flagpole. But what if we want to measure repeated occurrences of distance? Imagine, for example, a police car parked next to a warehouse. The rotating light from the police car would travel across the wall of the warehouse in regular intervals. If the input is time, the output would be the distance the beam of light travels. The beam of light would repeat the distance at regular intervals. The tangent function can be used to approximate this distance. Asymptotes would be needed to illustrate the repeated cycles when the beam runs parallel to the wall because, seemingly, the beam of light could appear to extend forever. The graph of the tangent function would clearly illustrate the repeated intervals. In this section, we will explore the graphs of the tangent and other trigonometric functions. Analyzing the graph of y = tan x We will begin with the graph of the tangent function, plotting points as we did for the sine and cosine functions. Recall that sin x ____ tan x = cos x The period of the tangent function is π because the graph repeats itself on intervals of kπ where k is a constant. If we π π __ __ graph the tangent function on   to  , w e can see the behavior of the graph on one complete cycle. If we look at any − 2 2 larger interval, we will see that the characteristics of the graph repeat. We can determine whether tangent is an odd or even function by using the definition of tangent. sin(−x) ______ tan(−x) = Definit ion of tangent. cos(−x) −sin x ______ = S ine is an odd function, cosine is even. cos x e q Th uo tient of an odd and an even sin x ____ = −  cos x function is odd. = −tan x Definit ion of tangent. e Th refore, tangent is an odd function. We can further analyze the graphical behavior of the tangent function by looking at values for some of the special angles, as listed in Table 1. π π π π π π π π _ _ _ _ _ _ _ _ −  −   −   −  0 x 2 3 4 6 6 4 3 2 — — — — √ 3 √ 3 ____ ____ −1 0 1 tan(x) undefined − √3 −     √ 3 undefined 3 3 Table 1 These points will help us draw our graph, but we need to determine how the graph behaves where it is undefined. π π π _ _ _ If we look more closely at values when x , we can use a table to look for a trend. Because ≈ 1.05 and 3 2 3 π _ ≈ 1.57, we will evaluate x at radian measures 1.05 x 1.57 as shown in Table 2. 2660 CHAPTER 8 Perio dic fu Nctio Ns x 1.3 1.5 1.55 1.56 tan x 3.6 14.1 48.1 92.6 Table 2 π _ As x approaches , the outputs of the function get larger and larger. Because y = tan x is an odd function, we see the 2 corresponding table of negative values in Table 3. −1.3 −1.5 −1.55 −1.56 x −3.6 −14.1 −48.1 −92.6 tan x Table 3 π _ We can see that, as x approaches −   , the outputs get smaller and smaller. Remember that there are some values of x 2 π 3π _ _ for which cos x = 0. For example, cos = 0 and cos = 0. At these values, the tangent function is undefined,     2 2 π 3π _ _ so the graph of y = tan x has discontinuities at x = and . A t these values, the graph of the tangent has vertical 2 2 π 3π _ _ asymptotes. Figure 1 represents the graph of y = tan(x). The tangent is positive from 0 to and from π to , 2 2 corresponding to quadrants I and III of the unit circle. y y = tan(x) 5 3 1 x –π π –1 –3 3π π π 3π x = − x = − x = x = 2 2 2 2 –5 Figure 1 Graph of the tangent function graphing variations of y = tan x As with the sine and cosine functions, the tangent function can be described by a general equation. y = Atan(Bx) We can identify horizontal and vertical stretches and compressions using values of A and B. The horizontal stretch can typically be determined from the period of the graph. With tangent graphs, it is often necessary to determine a vertical stretch using a point on the graph. Because there are no maximum or minimum values of a tangent function, the term amplitude cannot be interpreted as it is for the sine and cosine functions. Instead, we will use the phrase stretching/compressing factor when referring to the constant A. features of the graph of y = Atan(Bx) • e s Th tretching factor is A . ∣ ∣ π ___ • e p Th eriod is P = . B ∣ ∣ π π ____ _ • e do Th m ain is all real numbers x, where x ≠ + k such that k is an integer. 2 B ∣ ∣ B ∣ ∣ • e ra Th nge is ( −∞, ∞). π π _ _ • e a Th symptotes occur at x = + k, where k is an integer. 2 B B ∣ ∣ ∣ ∣ • y = Atan(Bx) is an odd function.