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Period and Frequency

Horizontal distance traveled before y values repeat; number of complete waves in 2pi.

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Changes in the Period of a Sine and Cosine Function

Your mission, should you choose to accept it, as Agent Trigonometry is to find the period of the cosine function y=cos[π(2x+4)].

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 2π. To stretch out the curve, then the period would have to be longer than 2π. Below we have sine curves with a period of 4π and then the second has a period of π.

To determine the period from an equation, we introduce b into the general equation. So, the equations are y=asinb(xh)+k and y=acosb(xh)+k, where a is the amplitude, b is the frequency, h is the phase shift, and k is the vertical shift. The frequency is the number of times the sine or cosine curve repeats within 2π. Therefore, the frequency and the period are indirectly related. For the first sine curve, there is half of a sine curve in 2π. Therefore the equation would be y=sin12x. The second sine curve has two curves within 2π, making the equation y=sin2x. To find the period of any sine or cosine function, use 2π|b|, where b is the frequency. Using the first graph above, this is a valid formula: 2π12=2π2=4π.

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

  1. y=3cos6x

The 6 in the equation tells us that there are 6 repetitions within 2π. So, the period is 2π6=π3.

  1. y=2sin14x

The 14 in the equation tells us the frequency. The period is 2π14=2π4=8π.

  1. y=sinπx7

The π is the frequency. The period is 2ππ=2.

Now, let's graph y=3cos6x  from [0,2π], determine where the maximum and minimum values occur, and state the domain and range.

The amplitude is -3, so it will be stretched and flipped. The period is π3 (from above) and the curve should repeat itself 6 times from 0 to 2π. The first maximum value is 3 and occurs at half the period, or x=π6 and then repeats at x=π2,5π6,7π6,3π2, Writing this as a formula we start at π6 and add π3 to get the next maximum, so each point would be (π6±π3n,3) where n is any integer.

The minimums occur at -3 and the x-values are multiples of π3. The points would be (±π3n,3), again n is any integer. The domain is all real numbers and the range is y[3,3].

Finally, let's find all the solutions from the function y=2sin14x  from [0,2π].

Now that the period can be different, we can have a different number of zeros within [0,2π]. In this case, we will have 6 times the number of zeros that the parent function. To solve this function, set y=0 and solve for x.

00=3cos6x=cos6x

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.

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

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

  1. Graph the function from \begin{align*}[0, 2\pi]\end{align*} and find the domain and range.
  2. Determine the coordinates of the maximum and minimum values.
  3. 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.

  1. Graph the function from \begin{align*}[0, 4\pi]\end{align*} and find the domain and range.
  2. Determine the coordinates of the maximum and minimum values.
  3. 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.

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

  1. Amplitude: -2 Period: \begin{align*}\frac{3 \pi}{4}\end{align*}
  2. Amplitude: \begin{align*}\frac{3}{5}\end{align*} Period: \begin{align*}5 \pi\end{align*}
  3. Amplitude: 9 Period: 6
  4. 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. 

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