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Graphing Sine and Cosine Help

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By McGraw-Hill Professional
Updated on Oct 4, 2011

Introduction to Graphing Sine and Cosine

The graph of a trigonometric function is a record of each cycle around the unit circle. For the function f ( x ) = sin x, x is the angle and f ( x ) is the y -coordinate of the terminal point determined by the angle x. In the function g (x) = cos x, g (x) is the x -coordinate of the terminal point determined by the angle x. For example, the point determined by the angle π /6 is , so f ( π /6) = sin π /6 = 1/2 and . We will sketch the graph of f ( x ) = sin x, using the points in Table 13.2.

Table 13.2

 x sin x Plot this point −2π sin(−2π) = 0 (−2π, 0) −3π/2 sin(−3π/2) = 1 (−3π/2, 1) −π sin(−π) = 0 (−π, 0) −π/2 sin(−π/2) = −1 (−π/2,−1) 0 sin 0 = 0 (0, 0) π/2 sin π/2 = 1 (π/2, 1) π sin π = 0 (π, 0) 3π/2 sin 3π/2 = −1 (3π/2,−1) 2π sin 2π = 0 (2π, 0)

Fig. 13.13

The graph in Figure 13.13 is two periods from the entire graph. This pattern repeats itself in both directions. Each period begins and ends at every multiple of 2 π : ..., [−2 π , 0], [0, 2 π ], [2 π , 4 π ], .... The graph between 0 and 2 π represents sine on the first positive cycle around the unit circle, between 2 π and 4 π represents the second positive cycle, and between 0 and −2 π represents the first negative cycle.

The graph for g ( x ) = cos x behaves in the same way. In fact, the graph of g ( x ) is the graph of f ( x ) shifted horizontally π /2 units. (We will see why this is true when we work with right triangles.) The graph for g ( x ) = cos x is shown in Figure 13.14.

Fig. 13.14

From their graphs, we can tell that f ( x ) = sin x is an odd function (sin(− x ) = − sin x ), and g ( x ) = cos x is even (cos(− x ) = cos x ). We can also see that their domain is all x and their range is all y values between −1 and 1.

The graphs of f ( x ) = sin x and g ( x ) = cos x can be shifted up or down, left or right, and stretched or compressed in the same way as other graphs. The graphs of y = c + sin x and y = c + cos x are shifted up or down c units. The graphs of y = a sin x and y = a cos x are vertically stretched or compressed, and the graphs of y = sin( xb ) and y = cos( xb ) are shifted horizontally by b units.

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