Graphing Sine and Cosine Functions

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, $y = \sin x$. We can create a table of values and use them to sketch a graph. Below are some of the values for the sine function on a unit circle, with the $x$-coordinate being the angle in radians and the $y$-coordinate being $\sin x$:

$\displaystyle{ (0, 0) \quad (\frac{\pi}{6}, \frac{1}{2}) \quad (\frac{\pi}{4}, \frac{\sqrt{2}}{2}) \quad (\frac{\pi}{3}, \frac{\sqrt{3}}{2}) \quad (\frac{\pi}{2}, 1) \\ (\frac{2\pi}{3}, \frac{\sqrt{3}}{2}) \quad (\frac{3\pi}{4}, \frac{\sqrt{2}}{2}) \quad (\frac{5\pi}{6}, \frac{1}{2}) \quad (\pi, 0) }$

Plotting the points from the table and continuing along the $x$-axis gives the shape of the sine function.

Graph of the sine function

Graph of points with $x$ coordinates being angles in radians, and $y$ coordinates being the function $\sin x$.

Notice how the sine values are positive between $0$ and $\pi$, 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 $\pi$ and $2\pi$, which correspond to the values of the sine function in quadrants III and IV on the unit circle.

Plotting values of the sine function

The points on the curve $y = \sin x$ correspond to the values of the sine function on the unit circle.

Now let’s take a similar look at the cosine function, $f(x) = \sin x$. Again, we can create a table of values and use them to sketch a graph. Below are some of the values for the sine function on a unit circle, with the $x$-coordinate being the angle in radians and the $y$-coordinate being $\cos x$:

$\displaystyle{ (0, 1) \quad (\frac{\pi}{6}, \frac{\sqrt{3}}{2}) \quad (\frac{\pi}{4}, \frac{\sqrt{2}}{2}) \quad (\frac{\pi}{3}, \frac{1}{2}) \quad (\frac{\pi}{2}, 0) \\ (\frac{2\pi}{3}, -\frac{1}{2}) \quad (\frac{3\pi}{4}, -\frac{\sqrt{2}}{2}) \quad (\frac{5\pi}{6}, -\frac{\sqrt{3}}{2}) \quad (\pi, -1) }$

As with the sine function, we can plots points to create a graph of the cosine function.

Graph of the cosine function

The points on the curve $y = \cos x$ correspond to the values of the cosine function on the unit circle.

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 $\left[-1, 1 \right]$.

Identifying Periodic Functions

In the graphs for both sine and cosine functions, the shape of the graph repeats after $2\pi$, which means the functions are periodic with a period of $2\pi$. A periodic function is a function with a repeated set of values at regular intervals. Specifically, it 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. The diagram below shows several periods of the sine and cosine functions.

Periods of the sine and cosine functions

The sine and cosine functions are periodic, meaning that a specific horizontal shift, $P$, results in a function equal to the original function:$f(x + P) = f(x)$.

Even and Odd Functions

Looking again at the sine and cosine functions on a domain centered at the $y$-axis helps reveal symmetries. As we can see in the graph of the sine function, it is symmetric about the origin, which indicates that it is an odd function. All along the graph, any two points with opposite $x$ values also have opposite $y$ values. This is characteristic of an odd function: two inputs that are opposites have outputs that are also opposites. In other words, if $\sin (-x) = - \sin x$.

Odd symmetry of the sine function

The sine function is odd, meaning it is symmetric about the origin.

The graph of the cosine function shows that it is symmetric about the y-axis. This indicates that it is an even function. For even functions, any two points with opposite $x$-values have the same function value. In other words, $\cos (-x) = \cos x$. We can see from the graph that this is true by comparing the $y$-values of the graph at any opposite values of $x$.

Even symmetry of the cosine function

The cosine function is even, meaning it is symmetric about the $y$-axis.