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The following graphs display the parent function for a $sin$, $cos$, and $tan$ function. Sine and cosine graphs are often referred to as waves, due to their pattern. The characteristics of these wave-like functions will be explored later.

As you can see, a sine curve looks like a wave. This means that it has certain wave-like properties such as an **amplitude** and a **period**.

The **amplitude** of a wave is the height of the wave from the $x$-axis to the peak. Note that the amplitude is the same for the peak above the axis and the peak below the axis. The amplitude of the example above, $y=sin(x)$, is $1$.

What the amplitude really tells us is the maximum and minimum value of the graph. This means that the maximum and minimum values of $y=sin(x)$ are $1$ and $−1$.

The **period** of a wave is the distance it takes to complete one wave-like pattern. For a sine graph, you can remember the pattern of the wave by the phrase “middle-high-middle-low-middle.” This suggests that the curve will start at the $x$-axis, increase to $y=1$, decrease to the $x$-axis again, and finish at $y=−1$.

In the example above, the period begins at $x=0$ and ends at $x=2π$. This means that the period is $2π$. The halfway point of the period is at $x=π$.

The amplitude of the parent cosine graph is the same as that of the sine graph: $1$. This means that the maximum value for a $y=cos(x)$ is $1$ and the minimum value is $−1$.

The period of the wave is also $2π$. However, the wave looks a bit different from the sine wave. The pattern for a cosine wave can be remembered as “high-middle-low-middle-high.” It begins at $y=1$ at the origin, decreases to $y=−1$, then increases back to $y=1$.

The graph of $y=tan(x)$ is much different from sine and cosine. As you can see, it is not represented by a wave like the others. Instead, you will see a cubic-like curve that repeats several times. Note that there are vertical asymptotes that separate each curve. If you need a refresher on asymptotes, review the chapter Nonlinear functions and graphs.

Beginning at the origin, you will see that the curves intersect the $x$-axis at intervals of $π$ in both the positive and negative $x$ directions. The vertical asymptotes occur at intervals of $π$ in both $x$ directions as well, but beginning at $x=π/2$.

You will often see sine and cosine functions that are a bit different from their parent functions. In these cases, we need to know how to find the amplitude and period without using a graph.

You can find the amplitude of a sine or cosine wave by finding the value in front of the function. The function $y=sin(x)$ has no number in front of the function, so we must assume that there is a $1$. That is why the amplitude of the functions above is $1$. Here is another example:

Find the amplitude of the following function:

$y=3∗sin(2x)$

The amplitude of the function is $3$. This is also the maximum value of the function. The minimum of the function is $−3$.

Find the period by figuring out what value you need to multiply into the inside of the function to get the coefficient of $x$ to equal $2π$. Normally, the inside of the function is just $x$. The coefficient here is $1$, and we have to multiply $1$ by $2π$ to get it to equal $2π$. Here is another example of finding the period:

Find the period of the following function:

$y=3∗sin(2x)$

Find out what value we need to multiply into the function to get a coefficient of $2π$ for $x$. The current coefficient is $2$.

$2∗period=2π$

The period of the function is $π$.

There are several trigonometric identities, which are the relationships between different trigonometric functions. However, for the ACT test you will only need to know one of them:

$sin_{2}(x)+cos_{2}(x)=1$

The $x$ in this identity can be anything, as long as the value is identical between the two functions. For instance, the functions could have $2x_{2}$ inside them instead of just $x$:

$sin_{2}(2x_{2})+cos_{2}(2x_{2})=1$

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