Your mission, should you choose to accept it, as Agent Trigonometry is to find the period of the cosine function

### Period

The last thing that we can manipulate on the sine and cosine curve is the **period**.

The normal period of a sine or cosine curve is

To determine the period from an equation, we introduce **frequency**,

Let's determine the period of the following sine and cosine functions.

y=−3cos6x

The 6 in the equation tells us that there are 6 repetitions within

y=2sin14x

The

y=sinπx−7

The

Now, let's graph

The amplitude is -3, so it will be stretched and flipped. The period is

The minimums occur at -3 and the

Finally, let's find all the solutions from the function

Now that the period can be different, we can have a different number of zeros within

Now, use the inverse cosine function to determine when the cosine is zero. This occurs at the multiples of \begin{align*}\frac{\pi}{2}\end{align*}.

\begin{align*}6x=\cos^{-1}0=\frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \frac{7\pi}{2},\frac{9\pi}{2}, \frac{11\pi}{2}, \frac{13\pi}{2}, \frac{15 \pi}{2}, \frac{17\pi}{2}, \frac{19\pi}{2}, \frac{21\pi}{2}, \frac{23\pi}{2}\end{align*}

We went much past \begin{align*}2 \pi\end{align*} because when we divide by 6, to get \begin{align*}x\end{align*} by itself, all of these answers are going to also be divided by 6 and smaller.

\begin{align*}x=\frac{\pi}{12}, \frac{\pi}{4}, \frac{5\pi}{12}, \frac{7\pi}{12}, \frac{3\pi}{4}, \frac{11\pi}{12}, \frac{13\pi}{12}, \frac{5\pi}{4}, \frac{17 \pi}{12}, \frac{19\pi}{12}, \frac{21\pi}{2}, \frac{23\pi}{12}\end{align*}

\begin{align*}\frac{23 \pi}{12}<2\pi\end{align*} so we have found all the zeros in the range.

### Examples

#### Example 1

Earlier, you were asked to find the period of \begin{align*}y=\cos [\pi (2x + 4)]\end{align*}.

First, we need to get the function in the form \begin{align*}y=a\cos b(x-h)+k\end{align*}. Therefore we need to factor out the 2.

\begin{align*}y=\cos [\pi (2x + 4)]\\ y = \cos [2\pi(x + 2)]\end{align*}

The \begin{align*}2\pi\end{align*} is the frequency. The period is therefore \begin{align*}\frac{2 \pi}{2\pi}=1\end{align*}.

#### Example 2

Determine the period of the function \begin{align*}y=\frac{2}{3}\cos\frac{3}{4}x\end{align*}.

The period is \begin{align*}\frac{2 \pi}{\frac{3}{4}}=2 \pi \cdot \frac{4}{3}=\frac{8 \pi}{3}\end{align*}.

#### Example 3

Find the zeros of the function from Example 2 from \begin{align*}[0, 2\pi]\end{align*}.

The zeros would be when \begin{align*}y\end{align*} is zero.

\begin{align*}0 &=\frac{2}{3} \cos \frac{3}{4}x \\ 0 &=\cos \frac{3}{4}x \\ \frac{3}{4}x &=\cos^{-1}0=\frac{\pi}{2},\frac{3 \pi}{2} \\ x &=\frac{4}{3}\left(\frac{\pi}{2},\frac{3 \pi}{2}\right) \\ x &=\frac{2\pi}{3},2\pi\end{align*}

#### Example 4

Determine the equation of the sine function with an amplitude of -3 and a period of \begin{align*}8\pi\end{align*}.

The general equation of a sine curve is \begin{align*}y=a\sin bx\end{align*}. We know that \begin{align*}a = -3\end{align*} and that the period is \begin{align*}8 \pi\end{align*}. Let’s use this to find the frequency, or \begin{align*}b\end{align*}.

\begin{align*}\frac{2\pi}{b} &=8\pi \\ \frac{2\pi}{8\pi} &=b \\ \frac{1}{4} &=b\end{align*}

The equation of the curve is \begin{align*}y=-3\sin \frac{1}{4}x\end{align*}.

### Review

Find the period of the following sine and cosine functions.

- \begin{align*}y=5\sin 3x\end{align*}
- \begin{align*}y=-2\cos 4x\end{align*}
- \begin{align*}y=-3\sin 2x\end{align*}
- \begin{align*}y=\cos \frac{3}{4}x\end{align*}
- \begin{align*}y=\frac{1}{2}\cos 2.5x\end{align*}
- \begin{align*}y=4\sin 3x\end{align*}

Use the equation \begin{align*}y=5\sin 3x\end{align*} to answer the following questions.

- Graph the function from \begin{align*}[0, 2\pi]\end{align*} and find the domain and range.
- Determine the coordinates of the maximum and minimum values.
- Find all the zeros from \begin{align*}[0, 2\pi]\end{align*}.

Use the equation \begin{align*}y=\cos \frac{3}{4}x\end{align*} to answer the following questions.

- Graph the function from \begin{align*}[0, 4\pi]\end{align*} and find the domain and range.
- Determine the coordinates of the maximum and minimum values.
- Find all the zeros from \begin{align*}[0, 2\pi]\end{align*}.

Use the equation \begin{align*}y=-3\sin 2x\end{align*} to answer the following questions.

- Graph the function from \begin{align*}[0, 2\pi]\end{align*} and find the domain and range.
- Determine the coordinates of the maximum and minimum values.
- Find all the zeros from \begin{align*}[0, 2\pi]\end{align*}.
- What is the domain of every sine and cosine function? Can you make a general rule for the range? If so, state it.

Write the equation of the sine function, in the form \begin{align*}y=a\sin bx\end{align*}, with the given amplitude and period.

- Amplitude: -2 Period: \begin{align*}\frac{3 \pi}{4}\end{align*}
- Amplitude: \begin{align*}\frac{3}{5}\end{align*} Period: \begin{align*}5 \pi\end{align*}
- Amplitude: 9 Period: 6
**Challenge**Find all the zeros from \begin{align*}[0, 2\pi]\end{align*} of \begin{align*}y=\frac{1}{2}\sin 3\left(x-\frac{\pi}{3}\right)\end{align*}.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 14.4.