Trigonometry Examples

y = - 2 \cot (\frac{π}{4} x)

$y = - 2 \cot (\frac{π}{4} x)$

Step 1

Use the form

a \cot (b x - c) + d

$a \cot (b x - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

a = - 2

$a = - 2$

b = \frac{π}{4}

$b = \frac{π}{4}$

c = 0

$c = 0$

d = 0

$d = 0$

Step 2

Since the graph of the function

c o t

$c o t$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Step 3

Find the period of

- 2 \cot (\frac{π x}{4})

$- 2 \cot (\frac{π x}{4})$ .

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Step 3.1

The period of the function can be calculated using

\frac{π}{| b |}

$\frac{π}{| b |}$ .

\frac{π}{| b |}

$\frac{π}{| b |}$

Step 3.2

Replace

b

$b$ with

\frac{π}{4}

$\frac{π}{4}$ in the formula for period.

\frac{π}{| \frac{π}{4} |}

$\frac{π}{| \frac{π}{4} |}$

Step 3.3

\frac{π}{4}

$\frac{π}{4}$ is approximately

0.78539816

$0.78539816$ which is positive so remove the absolute value

\frac{π}{\frac{π}{4}}

$\frac{π}{\frac{π}{4}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

π \frac{4}{π}

$π \frac{4}{π}$

Step 3.5

Cancel the common factor of

π

$π$ .

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Step 3.5.1

Cancel the common factor.

π \frac{4}{π}

$π \frac{4}{π}$

Step 3.5.2

Rewrite the expression.

4

$4$

4

$4$

4

$4$

Step 4

Find the phase shift using the formula

\frac{c}{b}

$\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from

\frac{c}{b}

$\frac{c}{b}$ .

Phase Shift:

\frac{c}{b}

$\frac{c}{b}$

Step 4.2

Replace the values of

c

$c$ and

b

$b$ in the equation for phase shift.

Phase Shift:

\frac{0}{\frac{π}{4}}

$\frac{0}{\frac{π}{4}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

0 (\frac{4}{π})

$0 (\frac{4}{π})$

Step 4.4

Multiply

0

$0$ by

\frac{4}{π}

$\frac{4}{π}$ .

Phase Shift:

0

$0$

Phase Shift:

0

$0$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period:

4

$4$

Phase Shift: None

Vertical Shift: None

Step 6