Trigonometry: Graphs: Sine and Cosine

To see how the sine and cosine functions are graphed, use a calculator, a computer, or a set of trigonometry tables to determine the values of the sine and cosine functions for a number of different degree (or radian) measures (see Table 1 ).

TABLE 1

Values of the Sine and Cosine at Various Angles

degrees	0°	30°	45°	60°	90°	120°
radians	0
sin x	0	0.500	0.707	0.866	1	0.866
cos x	1	0.866	0.707	0.500	0	−0.500
degrees	135°	150°	180°	210°	225°	240°
radians			π
sin x	0.707	0.500	0	−0.500	−0.707	−0.866
cos x	−0.707	−0.866	−1	−0.866	−0.707	−0.500

Next, plot these values and obtain the basic graphs of the sine and cosine function (Figure 1 ).

Figure 1

One period of the a) sine function and b) cosine function.

The sine function and the cosine function have periods of 2π; therefore, the patterns illustrated in Figure 1 are repeated to the left and right continuously (Figure 2 ).

Figure 2

Multiple periods of the a) sine function and b) cosine function.

Several additional terms and factors can be added to the sine and cosine functions, which modify their shapes.

The additional term A in the function y = A + sin x allows for a vertical shift in the graph of the sine functions. This also holds for the cosine function (Figure 3 ).

Figure 3

Examples of several vertical shifts of the sine function.

The additional factor B in the function y = B sin x allows for amplitude variation of the sine function. The amplitude, | B |, is the maximum deviation from the x-axis—that is, one half the difference between the maximum and minimum values of the graph. This also holds for the cosine function (Figure 4 ).

Figure 4

Examples of several amplitudes of the sine function.

Combining these figures yields the functions y = A + B sin x and also y = A + B cos x. These two functions have minimum and maximum values as defined by the following formulas. The maximum value of the function is M = A + |B|. This maximum value occurs whenever sin x = 1 or cos x = 1. The minimum value of the function is m = A - |B|. This minimum occurs whenever sin x = −1 or cos x = −1.