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# Math help from the Learning Centre

This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

## Trigonometric Functions

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.

## Sine Function

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:

## Cosine 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:

## Graphing Sinusoidal Functions

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:

Try this interactive tool!

Play around with the sliders for each value and observe what happens to both the $$sin(x)$$ and $$cos(x)$$ functions.