We set the units of the x axis to be degrees. The graph of y = sin x looks like a wave, as shown in Figure 1.12. (This is only part of the graph. The graph extends infinitely in both directions along the x axis.) For example, the point A = (1, x◦) corresponds to the point A1 = (x,sin x) on the curve y = sin x. If two points B1 and C1 are 360 from each other in the x direction, then they have the same y value, and they correspond to the same point B = C on the unit circle. (This is the correspondence of the identity sin(x◦+360◦) = sin x◦.)Also, the graph is symmetric about line x = 90. (This corresponds to the identity sin(90◦ −x◦) = sin(90◦ +x◦).) The identity sin(−x◦) = − sin x◦ indicates that the graph y = sin x is symmetric with respect to the origin; that is, the sine is an odd function. A function y = f (x) is sinusoidal if it can be written in the form y = f (x) = a sin[b(x + c)] + d for real constants a, b, c, and d. In particular, because cos x◦ = sin(x◦ + 90◦), y = cos x is sinusoidal (Figure 1.13). For any integer k, the graph of y = cos x is a (90 + 360k)-unit shift to the left (or a (270 + 360k)-unit shift to the right) of the graph of y = sin x. Because cos x◦ = cos(−x◦), the cosine is an even function, and so its graph is symmetric about the y axis. 1. Trigonometric Fundamentals 15 -250 -200 -150 -100 -50 50 100 150 200 1.2 1 0.8 0.6 0.4 0.2 -0.2 -0.4 -0.6 -0.8 -1 A A1 B B1 x -x x -x Figure 1.13. Example 1.5. Let f be an odd function defined on the real numbers such that for x ≥ 0, f (x) = 3 sin x + 4 cos x. Find f (x) for x < 0. (See Figure 1.14.) We leave it to the reader to show that if a, b, c, and d are real constants, then the functions y = a cos(bx+c)+d, y = a sin x+b cos x, y = a sin2 x, and y = b cos2 x are sinusoidal. Let f (x) and g(x) be two functions. For real constants a and b, the function af (x)+bg(x)is called a linear combination of f (x) and g(x). Is it true that if both of f (x) and g(x) are sinusoidal, then their linear combination is sinusoidal? In fact, it is true if f and g have the same period (or frequency). We leave this proof to the reader. Figure 1.16 shows the graphs of y1 = sin x, y3 = sin x + 1 3 sin 3x, and y5 = sin x + 1 3 sin 3x + 1 5 sin 5x. Can you see a pattern? Back in the nineteenth century, Fourier proved a number of interesting results, related to calculus, about the graphs of such functions yn as n goes to infinity. -250 -200 -150 -100 -50 50 100 150 200

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Expand and simplify (3x + 1)(x2 - 5x + 4). Tom and Samia are paid the same hourly rate.

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