Trigonometry Examples

$y = 3 \csc (\frac{θ}{4} + 5)$

Step 1

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

$a = 3$

$b = \frac{1}{4}$

$c = - 5$

$d = 0$

Step 2

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

Amplitude: None

Step 3

Find the period of $3 \csc (\frac{θ}{4} + 5)$ .

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Step 3.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2

Replace $b$ with $\frac{1}{4}$ in the formula for period.

$\frac{2 π}{| \frac{1}{4} |}$

Step 3.3

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{4}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 4$

Step 3.5

Multiply $4$ by $2$ .

$8 π$

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: $\frac{- 5}{\frac{1}{4}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- 5 \cdot 4$

Step 4.4

Multiply $- 5$ by $4$ .

Phase Shift: $- 20$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $8 π$

Phase Shift: $- 20$ ( $20$ to the left)

Vertical Shift: None

Step 6