The sine and cosine functions link the real numerical values of the x- and y-coordinates of the unit circle points. We write out functions in the form of:
\[ y = sin(x) \]
\[ y = cos(x) \]
In this section we will be discussing how to graph these two trigonometric functions, and the attributes that are responsible for how they appear.
Let us begin with the sine function. We can create a value table, and use several points to help us draw the graph. The table below lists some of the sine function values that can be found on the unit circle:
\(x\) (rad) | \( 0 \) | \( \frac{\pi}{6} \) | \( \frac{\pi}{4} \) | \( \frac{\pi}{3} \) | \( \frac{\pi}{2} \) | \( \frac{2\pi}{3} \) | \( \frac{3\pi}{4} \) | \( \frac{5\pi}{6} \) | \( \pi \) |
\(x\) (°) |
\( 0 \) | \( 30 \) | \( 45 \) | \( 60 \) | \( 90 \) | \( 120 \) | \( 135 \) | \( 150 \) | \( 180 \) |
\( sin(x) \) | \( 0 \) | \( \frac{1}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{\sqrt{3}}{2} \) | \(1 \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{1}{2} \) | \( 0 \) |
Plotting points from the table and continuing along the positive x-axis gives the shape of the sine function:
We can observe the cosine function in a similar way that we did the sine function. We can create a value table, and use several points to help us draw the graph. The table below lists some of the cosine function values that can be found on the unit circle:
\(x\) (rad) | \( 0 \) | \( \frac{\pi}{6} \) | \( \frac{\pi}{4} \) | \( \frac{\pi}{3} \) | \( \frac{\pi}{2} \) | \( \frac{2\pi}{3} \) | \( \frac{3\pi}{4} \) | \( \frac{5\pi}{6} \) | \( \pi \) |
\(x\) (°) |
\( 0 \) | \( 30 \) | \( 45 \) | \( 60 \) | \( 90 \) | \( 120 \) | \( 135 \) | \( 150 \) | \( 180 \) |
\( cos(x) \) | \( 1 \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{1}{2} \) | \(0 \) | \( \frac{-1}{2} \) | \( \frac{-\sqrt{2}}{2} \) | \( \frac{-\sqrt{3}}{2} \) | \( -1 \) |
Plotting the points from the table and continuing along the x-axis gives the shape of the cosine function:
A function that has the same general shape as a sine or cosine function is known as a sinusoidal function. The general forms of sinusoidal functions are:
where
Let's try some examples!
1. Graph the function \( y = 3sin( \frac{\pi}{4}x - \frac{\pi}{4}) \).
\( \Longrightarrow \) Let us find the amplitude of this function.
\( \Longrightarrow amplitude = |A| = |3| = 3\)
\( \Longrightarrow |A| > 1 \) and so we have a vertical stretch.
\( \Longrightarrow \) Let us determine the period.
\( \Longrightarrow period = \frac{2\pi}{|B|} = \frac{2\pi}{|\frac{\pi}{4}|} = 2\pi \times \frac{4}{\pi} = 8 \)
\( \Longrightarrow |B| < 1 \) and so we have a horizontal stretch.
\( \Longrightarrow \) Lastly, we determine the phase shifts. Since there is no "D" value, we only consider a horizontal shift.
\( \Longrightarrow \frac{C}{B} = \frac{\frac{\pi}{4}}{\frac{\pi}{4}} = 1 \)
\( \Longrightarrow \frac{C}{B} > 0 \) and so we have a horizontal shift of 1 unit to the right.
Before we graph our function, we need to find our \( y-intercept \), and a proper graph will show at least one full period (which is done by adding \( \frac{1}{4} \) of the period to our x-value four times. See the image below for the final result:
2. Graph the function \( y = -2cos(\frac{\pi}{2}x + \pi) + 3 \).
See the video below for the solution: