52 Graphs of the Sine and Cosine Functions
Learning Objectives
In this section, you will:
- Graph variations of y=sin( x ) and y=cos( x ).
- Use phase shifts of sine and cosine curves.
White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can be seen only when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow.
Light waves can be represented graphically by the sine function. In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions.
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. Let’s start with the sine function. We can create a table of values and use them to sketch a graph. (Figure) lists some of the values for the sine function on a unit circle.
| [latex]x[/latex] | [latex]0[/latex] | [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]\mathrm{sin}\left(x\right)[/latex] | [latex]0[/latex] | [latex]\frac{1}{2}[/latex] | [latex]\frac{\sqrt{2}}{2}[/latex] | [latex]\frac{\sqrt{3}}{2}[/latex] | [latex]1[/latex] | [latex]\frac{\sqrt{3}}{2}[/latex] | [latex]\frac{\sqrt{2}}{2}[/latex] | [latex]\frac{1}{2}[/latex] | [latex]0[/latex] |
Plotting the points from the table and continuing along the x-axis gives the shape of the sine function. See (Figure).
Notice how the sine values are positive between 0 and[latex]\,\pi ,\,[/latex]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[latex]\,\pi \,[/latex]and[latex]\,2\pi ,\,[/latex]which correspond to the values of the sine function in quadrants III and IV on the unit circle. See (Figure).
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. (Figure) lists some of the values for the cosine function on a unit circle.
| [latex]\mathbf{x}[/latex] | [latex]0[/latex] | [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]\mathbf{cos}\left(\mathbf{x}\right)[/latex] | [latex]1[/latex] | [latex]\frac{\sqrt{3}}{2}[/latex] | [latex]\frac{\sqrt{2}}{2}[/latex] | [latex]\frac{1}{2}[/latex] | [latex]0[/latex] | [latex]-\frac{1}{2}[/latex] | [latex]-\frac{\sqrt{2}}{2}[/latex] | [latex]-\frac{\sqrt{3}}{2}[/latex] | [latex]-1[/latex] |
As