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# Tangent Graphs

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Practice Tangent Graphs
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Changes in the Period of a Sine and Cosine Function

What is the period of the cosine function $y=\cos [\pi (2x + 4)]$ ?

### Guidance

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

To determine the period from an equation, we introduce $b$ into the general equation. So, the equations are $y=a\sin b(x-h)+k$ and $y=a\cos b(x-h)+k$ , 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 $2 \pi$ . Therefore, the frequency and the period are indirectly related. For the first sine curve, there is half of a sine curve in $2 \pi$ . Therefore the equation would be $y=\sin \frac{1}{2}x$ . The second sine curve has two curves within $2 \pi$ , making the equation $y=\sin 2x$ . To find the period of any sine or cosine function, use $\frac{2 \pi}{|b|}$ , where $b$ is the frequency. Using the first graph above, this is a valid formula: $\frac{2 \pi}{\frac{1}{2}}=2 \pi \cdot 2=4 \pi$ .

#### Example A

Determine the period of the following sine and cosine functions.

a) $y=-3 \cos 6x$

b) $y=2 \sin \frac{1}{4}x$

c) $y=\sin \pi x -7$

Solution: a) The 6 in the equation tells us that there are 6 repetitions within $2 \pi$ . So, the period is $\frac{2 \pi}{6}=\frac{\pi}{3}$ .

b) The $\frac{1}{4}$ in the equation tells us the frequency. The period is $\frac{2 \pi}{\frac{1}{4}}=2 \pi \cdot 4=8 \pi$ .

c) The $\pi$ is the frequency. The period is $\frac{2 \pi}{\pi}=2$ .

#### Example B

Graph part a) from the previous example from $[0, 2 \pi]$ . Determine where the maximum and minimum values occur. Then, state the domain and range.

Solution: The amplitude is -3, so it will be stretched and flipped. The period is $\frac{\pi}{3}$ (from above) and the curve should repeat itself 6 times from 0 to $2 \pi$ . The first maximum value is 3 and occurs at half the period, or $x=\frac{\pi}{6}$ and then repeats at $x=\frac{\pi}{2}, \frac{5 \pi}{6}, \frac{7 \pi}{6}, \frac{3 \pi}{2}, \ldots$ Writing this as a formula we start at $\frac{\pi}{6}$ and add $\frac{\pi}{3}$ to get the next maximum, so each point would be $\left(\frac{\pi}{6} \pm \frac{\pi}{3}n,3\right)$ where $n$ is any integer.

The minimums occur at -3 and the $x$ -values are multiples of $\frac{\pi}{3}$ . The points would be $\left(\pm \frac{\pi}{3}n, -3\right)$ , again $n$ is any integer. The domain is all real numbers and the range is $y \in [-3,3]$ .

#### Example C

Find all the solutions from the function in Example B from $[0, 2 \pi]$ .

Solution: Before this concept, the zeros didn’t change in the frequency because we hadn’t changed the period. Now that the period can be different, we can have a different number of zeros within $[0, 2\pi]$ . 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 $x$ .

$0 &=-3 \cos 6x \\0 &=\cos 6x$

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

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

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

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

$\frac{23 \pi}{12}<2\pi$ so we have found all the zeros in the range.

Concept Problem Revisit First, we need to get the function in the form $y=a\cos b(x-h)+k$ . Therefore we need to factor out the 2.

$y=\cos [\pi (2x + 4)]\\y = \cos [2\pi(x + 2)]$

The $2\pi$ is the frequency. The period is therefore $\frac{2 \pi}{2\pi}=1$ .

### Guided Practice

1. Determine the period of the function $y=\frac{2}{3}\cos\frac{3}{4}x$ .

2. Find the zeros of the function from #1 from $[0, 2\pi]$ .

3. Determine the equation of the sine function with an amplitude of -3 and a period of $8\pi$ .

1. The period is $\frac{2 \pi}{\frac{3}{4}}=2 \pi \cdot \frac{4}{3}=\frac{8 \pi}{3}$ .

2. The zeros would be when $y$ is zero.

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

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

$\frac{2\pi}{b} &=8\pi \\\frac{2\pi}{8\pi} &=b \\\frac{1}{4} &=b$

The equation of the curve is $y=-3\sin \frac{1}{4}x$ .

### Vocabulary

Period
The length in which an entire sine or cosine curve is completed.
Frequency
The number of times a curve is repeated within $2\pi$ .

### Practice

Find the period of the following sine and cosine functions.

1. $y=5\sin 3x$
2. $y=-2\cos 4x$
3. $y=-3\sin 2x$
4. $y=\cos \frac{3}{4}x$
5. $y=\frac{1}{2}\cos 2.5x$
6. $y=4\sin 3x$

Use the equation $y=5\sin 3x$ to answer the following questions.

1. Graph the function from $[0, 2\pi]$ and find the domain and range.
2. Determine the coordinates of the maximum and minimum values.
3. Find all the zeros from $[0, 2\pi]$ .

Use the equation $y=\cos \frac{3}{4}x$ to answer the following questions.

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

Use the equation $y=-3\sin 2x$ to answer the following questions.

1. Graph the function from $[0, 2\pi]$ and find the domain and range.
2. Determine the coordinates of the maximum and minimum values.
3. Find all the zeros from $[0, 2\pi]$ .
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 $y=a\sin bx$ , with the given amplitude and period.

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