Trigonometry Examples

$f (x) = 4 \cot (\frac{5}{8} x)$

Step 1

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

$a = 4$

$b = \frac{5}{8}$

$c = 0$

$d = 0$

Step 2

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

Amplitude: None

Step 3

Find the period of $4 \cot (\frac{5 x}{8})$ .

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Step 3.1

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Step 3.2

Replace $b$ with $\frac{5}{8}$ in the formula for period.

$\frac{π}{| \frac{5}{8} |}$

Step 3.3

$\frac{5}{8}$ is approximately $0.625$ which is positive so remove the absolute value

$\frac{π}{\frac{5}{8}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$π \frac{8}{5}$

Step 3.5

Combine $π$ and $\frac{8}{5}$ .

$\frac{π \cdot 8}{5}$

Step 3.6

Move $8$ to the left of $π$ .

$\frac{8 π}{5}$

Step 4

Find the phase shift using the formula $\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 4.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{0}{\frac{5}{8}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{8}{5})$

Step 4.4

Multiply $0$ by $\frac{8}{5}$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{8 π}{5}$

Phase Shift: None

Vertical Shift: None

Step 6