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

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

** Figure 1
**

The sine function and the cosine function have periods of 2π; therefore, the patterns illustrated in Figure

**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.

**Example 1:** Graph the function *y* = 1 + 2 sin *x*. What are the maximum and minimum values of the function?

The maximum value is 1 + 2 = 3. The minimum value is 1 −2 = −1 (Figure 5

** Figure 5**

Drawing for Example 1.

**Example 2:** Graph the function *y* = 4 + 3 sin *x*. What are the maximum and minimum values of the function?

The maximum value is 4 + 3 = 7. The minimum value is 4 − 3 = 1 (Figure 6

** Figure 6**

Drawing for Example 2.

The additional factor *C* in the function *y* = sin *Cx* allows for **period** variation (length of cycle) of the sine function. (This also holds for the cosine function.) The period of the function *y* = sin *Cx* is 2π/|C|. Thus, the function *y* = sin 5 *x* has a period of 2π/5. Figure 7

** Figure 7**

Examples of several frequencies of the a) sine function and b) cosine function.

The additional term *D* in the function *y* = sin ( *x* + *D*) allows for a **phase shift** (moving the graph to the left or right) in the graph of the sine functions. (This also holds for the cosine function.) The phase shift is | *D* |. This is a positive number. It does not matter whether the shift is to the left (if *D* is positive) or to the right (if *D* is negative). The sine function is odd, and the cosine function is even. The cosine function looks exactly like the sine function, except that it is shifted π/2 units to the left (Figure 8

** Figure 8**

Examples of several phase shifts of the sine function.

**Example 3:** What is the amplitude, period, phase shift, maximum, and minimum values of

*y* = 3+2 sin (3 *x*‐2)

*y* = 4 cos2π *x*

**Example 4:** Sketch the graph of *y* = cosπ *x*.

Because cos *x* has a period of 2π, cos π *x* has a period of 2 (Figure 9

** Figure 9
** Drawing for Example 4.

**Example 5:** Sketch the graph of *y* = 3 cos (2x + π/2).

Because cos *x* has a period of 2π, cos 2x has a period of π (Figure 10

**Figure 10**

Drawing for Example 5.

The graph of the function *y* = − *f*( *x*) is found by reflecting the graph of the function *y* = *f*( *x*) about the *x*‐axis. Thus, Figure *y* = −3 sin 2 *x*. Specifically,

It is important to understand the relationships between the sine and cosine functions and how phase shifts can alter their graphs.