Trigonometry Examples

$y = \csc (\frac{1}{2} x - \frac{π}{4})$

Step 1

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

$a = 1$

$b = \frac{1}{2}$

$c = \frac{π}{4}$

$d = 0$

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 $\csc (\frac{x}{2} - \frac{π}{4})$ .

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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 $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

Step 3.3

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Step 3.5

Multiply $2$ by $2$ .

$4 π$

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: $\frac{\frac{π}{4}}{\frac{1}{2}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{4} \cdot 2$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Factor $2$ out of $4$ .

Phase Shift: $\frac{π}{2 (2)} \cdot 2$

Step 4.4.2

Cancel the common factor.

Phase Shift: $\frac{π}{2 \cdot 2} \cdot 2$

Step 4.4.3

Rewrite the expression.

Phase Shift: $\frac{π}{2}$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $4 π$

Phase Shift: $\frac{π}{2}$ ( $\frac{π}{2}$ to the right)

Vertical Shift: None

Step 6