VIEWS: 19 PAGES: 31 POSTED ON: 10/3/2012 Public Domain
8.2 Sine and Cosine Curves Objective To find equations of different sine and cosine curves and to apply these equations. Sine and Cosine Curves In section 4.4, we have learned that the graph of y = cf(x) can be obtained by vertically stretching or shrinking and even reflecting the graph of y = f(x) . Sine and Cosine Curves Also in section 4.4, we learned that the graph of y = f(cx) can be obtained by horizontally stretching or shrinking the graph of y = f(x) . In general, we can determine useful information about the graphs of y = AsinBx and y = AcosBx by analyzing the factors of A and B. Before we further discuss the graphs of sine and cosine, let’s review the properties of sine and cosine. Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 1. The domain is the set of all real numbers. 2. The range is the set of y values such that –1 y 1. 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 5. Each function cycles through all the values of the range over an x-interval of 2. 6. The cycle repeats itself indefinitely in both directions of the x-axis. Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 3 x 0 2 2 2 sin x 0 1 0 -1 0 Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y y = sin x 3 1 3 5 2 2 2 2 2 2 x 1 Remember that y = sin x is an odd function whose graph is symmetry to origin. Here is the graph y = f(x) = sin x showing from -2 to 6. Notice it repeats with a period of 2. 2 2 2 2 It has a maximum of 1 and a minimum of -1 (remember that is the range of the sine function) From Algebra recall that an odd function (which the sine is) is symmetric with respect to the origin as can be seen here What are the x intercepts? Where does sin x = 0? …-3, -2, -, 0, , 2, 3, 4, . . . 7 3 5 2 2 2 2 3 2 0 2 3 4 Where is the function maximum? Where does sin x = 1? 7 3 5 , , , 2 2 2 2 Where is the function minimum? Where does sin x = -1? 5 3 7 , , , 2 2 2 2 7 3 5 2 2 2 2 3 2 0 2 3 4 5 3 7 2 2 2 2 Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 3 x 0 2 2 2 cos x 1 0 -1 0 1 Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y y = cos x 3 1 3 5 2 2 2 2 2 2 x 1 Remember that y = cos x is an even function whose graph is symmetry to y-axis. Here is the graph y = f(x) = cosx showing from -2 to 6. Notice it repeats with a period of 2. 2 2 2 2 It has a maximum of 1 and a minimum of -1 (remember that is the range of the cosine function) Recall that an even function (which the cosine is) is symmetric with respect to the y axis as can be seen here What are the x intercepts? Where does cos x = 0? 3 3 5 , , , , 2 2 2 2 2 2 0 2 3 3 5 2 2 2 2 2 Where is the function maximum? Where does cos x = 1? …-4, -2, 0, 2, 4, . . . Where is the function minimum? Where does cos x = -1? …-3, -, , 3, . . . 2 0 2 3 3 5 2 4 2 2 2 2 3 3 What would happen if we multiply the function by a constant? All function values would be twice as high y = 2 sin x amplitude is here amplitude of this y = 2 sin x graph is 2 y = sin x The highest the graph goes (without a vertical shift) is called the amplitude. Example: Sketch the graph of y = 3cos x on the interval [–, 4]. Partition the interval [0, 2] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. 3 0 2 2 2 x 3 0 -3 0 3 y = cosx 1 0 -1 0 1 y = 3cosx 3 0 -3 0 3 max x-int min x-int max (0, 3) y (2, 3) 2 1 2 3 4 x 1 ( 3 , 0) 2 ( , 0) 2 2 3 ( , –3) The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically. If 0 < |a| < 1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis. y 4 y = 2sin x 3 2 2 2 x 1 y= 2 sin x y = – 4 sin x y = sin x reflection of y = 4 sin x y = 4 sin x 4 For y = A cos x and y = A sin x, A is the amplitude. What is the amplitude for the following? y = 4 cos x y = -3 sin x amplitude is 4 amplitude is 3 The last thing we want to see is what happens if we put a coefficient on the x. y = sin 2x y = sin 2x y = sin x It makes the graph "cycle" twice as fast. It does one complete cycle in half the time so the period becomes . The period of a function is the x interval needed for the function to complete one cycle. For b 0, the period of y = a sinbx is 2 . b For b 0, the period of y = a cosbx is also 2 . b If b > 1, the graph of the function is shrunk horizontally. y y sin 2 x period: 2 period: y sin x x 2 If 0 < b < 1, the graph of the function is stretched horizontally. y y cos x 1 y cos x period: 2 2 2 3 4 x period: 4 For b < 0, what is the period of y = a sinbx and y = a cosbx ? Use basic odd-even properties to the trigonometric function sin (–u) = – sinu cos(–u) = cosu The period of y = 3sin (–4x) is the same period as y = – 3sin4x. So the period of y = 3sin (–4x) is 2 The period of y = 2cos (–5x) is the same period as y = 2cos5x. 2 So the period of y = 2cos (–5x) is 5 So if we look at y = sinx the affects the period. This will be 2 true for The period T = cosine as well. 2 What is the period of y = cos4x? T 4 2 y = cos x This means the graph will "cycle" every /2 or 4 times as often y = cos 4x What do you think will happen to the graph if we put a fraction in front? 1 y sin x 2 y = sin 1/2 x y = sin x The period for one complete cycle is twice as long or 4 absolute value of this is the amplitude y A cost y A sin t Period is 2 divided by absolute value of this Use basic trigonometric identities to graph y = f (–x) Example 1: Sketch the graph of y = sin (–x). The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. y y = sin (–x) Use the identity sin (–x) = – sin x x y = sin x 2 Example 2: Sketch the graph of y = cos (–x). The graph of y = cos (–x) is identical to the graph of y = cos x. y Use the identity cos (–x) = cos x x 2 y = cos (–x) Example 3: Sketch the graph of y = 2sin (–3x). Rewrite the function in the form y = a sin bx with b > 0 Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x amplitude: |a| = |–2| = 2 period: 2 = 2 b 3 Partition the [0, 2/3] into 4 Calculate the five key points. subintervals. x 2 0 6 3 2 3 y = –2 sin 3x 0 –2 0 2 0 y ( , 2) 2 2 2 5 6 6 3 2 3 6 x (0, 0) ( , 0) 2 2 3 ( , 0) ( , -2) 3 6 Practice 1. Give the amplitude and period of the function y 4sin 3x. Then sketch at least one cycle of its graph. 2 2 amplitude = 4 4 period = B 3 Key points: x sin3x x 4sin 3x x 4sin3x 0 0 0 0 0 0 1 6 4 4 6 6 3 0 3 0 3 0 2 1 2 4 4 2 2 2 0 2 3 0 3 3 0 Practice 2. Give the amplitude, period, and an equation of the curve shown. Use the cosine curve, y A cos Bx, with amplitude 3 and period 8. 2 2 8 B B 8 4 So the equation is y 3cos x. 4 Practice 3. Solve the equation 6sin 2 x 5 for 0 x 2 . Give the answer to the nearest hundredth of a radian 5 6sin 2x 5 sin 2 x 6 2 x .99, 2.16, 7.27, 8.44 If 0 x 2 , then 0 2 x 4 x 0.49, 1.08, 3.64, 4.22 or graph y1 6sin 2 x and y2 5 and find the intersections over the interval 0 x 2 Assignment P. 305 #1 – 10, 11 – 16, 21 – 23