Graphs of the Sine and Cosine Function: Learn It 1

  • Determine amplitude, period, phase shift, and vertical shift of a sine or cosine graph from its equation.
  • Graph transformations of y=cos x and y=sin x .
  • Determine a function formula that would have a given sinusoidal graph.
  • Determine functions that model circular and periodic motion.

The Standard Graph of [latex]y=sin(x)[/latex]  and [latex]y=cos(x)[/latex]

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 [latex]\frac{\pi}{6}[/latex] [latex]\frac{\pi}{4}[/latex] [latex]\frac{\pi}{3}[/latex] [latex]\frac{\pi}{2}[/latex] [latex]\frac{2\pi}{3}[/latex] [latex]\frac{3\pi}{4}[/latex] [latex]\frac{5\pi}{6}[/latex] [latex]\pi[/latex]
[latex]\sin(x)[/latex] 0 [latex]\frac{1}{2}[/latex] [latex]\frac{\sqrt{2}}{2}[/latex] [latex]\frac{\sqrt{3}}{2}[/latex] 1 [latex]\frac{\sqrt{3}}{2}[/latex] [latex]\frac{\sqrt{2}}{2}[/latex] [latex]\frac{1}{2}[/latex] 0

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

A graph of sin(x). Local maximum at (pi/2, 1). Local minimum at (3pi/2, -1). Period of 2pi.
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.

A side-by-side graph of a unit circle and a graph of sin(x). The two graphs show the equivalence of the coordinates.
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 [latex]\frac{\pi}{6}[/latex] [latex]\frac{\pi}{4}[/latex] [latex]\frac{\pi}{3}[/latex] [latex]\frac{\pi}{2}[/latex] [latex]\frac{2\pi}{3}[/latex] [latex]\frac{3\pi}{4}[/latex] [latex]\frac{5\pi}{6}[/latex] [latex]\pi[/latex]
[latex]\cos(x)[/latex] 1 [latex]\frac{\sqrt{3}}{2}[/latex] [latex]\frac{\sqrt{2}}{2}[/latex] [latex]\frac{1}{2}[/latex] 0 [latex]-\frac{1}{2}[/latex] [latex]-\frac{\sqrt{2}}{2}[/latex] [latex]-\frac{\sqrt{3}}{2}[/latex] −1

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

A graph of cos(x). Local maxima at (0,1) and (2pi, 1). Local minimum at (pi, -1). Period of 2pi.
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 [latex]2\pi[/latex], which means the functions are periodic with a period of [latex]2pi[/latex].

period of a function

The period of a trigonometric function is the length of one complete cycle of the function’s graph

Side-by-side graphs of sin(x) and cos(x). Graphs show period lengths for both functions, which is 2pi.

Symmetry of Sine and Cosine

The sine function is an odd function because [latex]\sin(−x)=−\sin x[/latex].

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.
Odd symmetry of the sine function

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 [latex]\cos(−x)=\cos x[/latex].

 

A graph of cos(x) that shows that cos(x) is an even function due to the even symmetry of the graph.
Even symmetry of the cosine function

characteristics of sine and cosine graphs

The sine and cosine functions have several distinct characteristics:

  • They are periodic functions with a period of 2π.
  • The domain of each function is [latex]\left(-\infty,\infty\right)[/latex] and the range is [latex]\left[−1,1\right][/latex].
  • The graph of [latex]y=\sin x[/latex] is symmetric about the origin, because it is an odd function.
  • The graph of [latex]y=\cos x[/latex] is symmetric about the y-axis, because it is an even function.