EXAMPLE
for Exs. 3-14
ANALYZING FUNCTIONS Identify the amplitude and the period of the graph of
the function
Answers
Step-by-step explanation:
Cosine Function
Graph variations of y=sin( x ) and y=cos( x )
Recall that the sine and cosine functions relate real number values to the x– and y-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let’s start with the sine function. We can create a table of values and use them to sketch a graph. The table below lists some of the values for the sine function on a unit circle.
x 0
π
6
π
6
π
3
π
2
2
π
3
3
π
4
5
π
6
π
sin(x) 0
1
2
√
2
2
√
3
2
1
√
3
2
√
2
2
1
2
0
Plotting the points from the table and continuing along the x-axis gives the shape of the sine function. See Figure 2.
A graph of sin(x). Local maximum at (pi/2, 1). Local minimum at (3pi/2, -1). Period of 2pi.
Figure 2. The sine function
Notice how the sine values are positive between 0 and π, which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between π and 2π, which correspond to the values of the sine function in quadrants III and IV on the unit circle. See Figure 3.
A side-by-side graph of a unit circle and a graph of sin(x). The two graphs show the equivalence of the coordinates.
Figure 3. Plotting values of the sine function
Now let’s take a similar look at the cosine function. Again, we can create a table of values and use them to sketch a graph. The table below lists some of the values for the cosine function on a unit circle.
x 0
π
6
π
4
π
3
π
2
2
π
3
3
π
4
5
π
6
π
cos(x) 1
√
3
2
√
2
2
1
2
0
−
1
2
−
√
2
2
−
√
3
2
−1
As with the sine function, we can plots points to create a graph of the cosine function as in Figure 4.
A graph of cos(x). Local maxima at (0,1) and (2pi, 1). Local minimum at (pi, -1). Period of 2pi.
Figure 4. The cosine function
Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval [−1,1].
In both graphs, the shape of the graph repeats after 2π,which means the functions are periodic with a period of 2π. A periodic function is a function for which a specific horizontal shift, P, results in a function equal to the original function: f (x + P) = f (x) for all values of x in the domain of f. When this occurs, we call the smallest such horizontal shift with P > 0 the period of the function. Figure 5 shows several periods of the sine and cosine functions.
Side-by-side graphs of sin(x) and cos(x). Graphs show period lengths for both functions, which is 2pi.
Figure 5
Looking again at the sine and cosine functions on a domain centered at the y-axis helps reveal symmetries. As we can see in Figure 6, the sine function is symmetric about the origin. Recall from The Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because
s
i
n
(
−
x
)
=
−
s
i
n
x
. Now we can clearly see this property from the graph.
A graph of sin(x) that shows that sin(x) is an odd function due to the odd symmetry of the graph.
Figure 6. Odd symmetry of the sine function
Figure 7 shows that the cosine function is symmetric about the y-axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that
cos
(
−
x
)
=
cos
x
.