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# Sine Function

You've probably studied a variety of functions by now, one of the most important in trigonometry as the sine function. The sine function is a periodic function, meaning that its graph behaves cyclically by repeating itself over and over from left to right. The graph of $y=\mathrm{sin}\left(x\right)$ is below for reference:

If that doesn't make much sense, don't worry about it. This article will look at the sine function in other ways to make it a more useful tool in your mathematics repertoire. Let's get started!

## Understanding the sine function

The easiest way to examine the sine function is to use the unit circle. For any given angle measure θ, we can draw a unit circle on the coordinate plane with the angle centered at the origin and the positive x-axis as one side. This is what it should look like:

The y-coordinate of the point on the circle is $\mathrm{sin}\left(\theta \right)$ , and the x-coordinate is $\mathrm{cos}\left(\theta \right)$ .

## Using the sine function to memorize important values

While we don't need to use the sine function to illustrate important sine values, having a visual can only help. The following diagram highlights the properties of 45-45-90 right triangles using the sine function:

If you haven't memorized this information yet, seeing it like this may help you. Similarly, the diagrams below illustrate important sine values for 30-60-90 right triangles:

The true value of the sine function is revealed when we start using the values above to calculate other values. The diagram below provides a lot of information in a relatively small space:

Remember that the value of sin(θ) will be positive in quadrants I and II and negative in quadrants III and IV.

## Deepening our understanding of the sine function

The period of $f\left(x\right)=\mathrm{sin}\left(\theta \right)$ is $2\pi$ , meaning that it starts repeating at every interval of $2\pi$ . Therefore, we can see the whole graph by sketching it on the coordinate plane with x values from 0 to $2\pi$ :

If we extended the x-axis beyond $2\pi$ , the curve would repeat the pattern above. Therefore, the image above is the "standard" depiction of the sine function when graphed as a line. Put another way, the domain of the sine function is all real numbers while the range is $-1\le y\le 1$ .

Unlike parabolas, the sine function doesn't have a vertex with a clear highest or lowest point. Since the highest and lowest points repeat every ` $2\pi$ ` tough, we can express both algebraically:

Highest point: $x=\frac{\pi }{2}+2n\pi$ , where n must be an integer

Lowest point: $x=\frac{3\pi }{2}+2n\pi$ , where n must be an integer

## Modifying the sine function

While the rules above are true for the basic sine function $f\left(x\right)=\mathrm{sin}\left(\theta \right)$ , things change slightly if we're working with more complex functions. The standard form of a sine function is:

$y=a\cdot \mathrm{sin}\left(bx\right)$

We use the a value to determine the amplitude of the function, or how far it goes above and below the x-axis. The amplitude doesn't change the x-intercepts of the graph, just what the y values are. The diagram below illustrates this by comparing the graph of f $\left(x\right)=\mathrm{sin}\left(x\right)>$ $f\left(x\right)=2×\mathrm{sin}\left(x\right)$ :

The domain of the red graph above is still all real numbers, but the range has increased to $-2\le y\le 2$ . Importantly, it doesn't matter if a is positive or negative. The absolute value of 'a' ( $|a|$ ) determines the amplitude. Put another way, $y=-2\mathrm{sin}\left(x\right)$ and $y=2\mathrm{sin}\left(x\right)$ have the same amplitude.

Likewise, the 'b' value affects the period of the sine function, or how long it takes to repeat. The formula is:

$\mathrm{Period}=\frac{2\pi }{|b|}$

The 'b' value of $y=\mathrm{sin}\left(x\right)$ and $y=2\mathrm{sin}\left(x\right)$ is 1, which means that both functions have the same period. If we substitute 1 into the formula above, we get:

$\frac{2\pi }{|1|}$

That works out to $2\pi$ , which is the period of the standard sine function. Note that we're using absolute value again, so 'b' can be positive or negative without changing the period.

## Practice questions on sine function

a. Find the amplitude of the following sine function: $y=\frac{-3}{\mathrm{sin}\left(x\right)}$

The amplitude is the absolute value of a when the function is written in standard form. The absolute value of this function is 3, making 3 our amplitude.

b. Define the domain and range of the following sine function: $y=\frac{4}{\mathrm{sin}\left(x\right)}$

The amplitude of a sine function determines its range, so we need to start with that. The a value is 4 and the absolute value of 4 is still 4, so we have our amplitude. That means that the range of this function is $-4\le y\le 4$ . The domain is x can be all real numbers for any sine function.

c. Find the period of the following sine function: $f\left(x\right)=\mathrm{sin}\left(-4x\right)$

The formula for determining the period of a sine function is $\frac{2\pi }{|b|}$ . Our b value is -4, so we take the absolute value of that and put it under the $2\pi$ . That gives us:

$\frac{2\pi }{4}$

That works out to $\frac{1}{2\pi }$ , meaning that the pattern repeats every $\frac{1}{2\pi }$ .

d. Find the amplitude and period of the following sine function: $f\left(x\right)=-3\mathrm{sin}\left(2x\right)$

This problem gives us a and b values to work with. Let's begin with the amplitude. The a value is -3, so we take the absolute value of that for an amplitude of 3.

Next, we take the absolute value of b (2) and plug it into our period formula:

$\frac{2\pi }{|b|}$

$\frac{2\pi }{|2|}$

That works out to $\pi$ . Therefore, we've successfully solved the problem. The amplitude is 3, and the period is $\pi$ .

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