Trigonometry Examples

$y = 4 \sin (\frac{2 π x}{5} - \frac{5}{4})$

Step 1

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

$a = 4$

$b = \frac{2 π}{5}$

$c = \frac{5}{4}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $4$

Step 3

Find the period of $4 \sin (\frac{2 π x}{5} - \frac{5}{4})$ .

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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 $\frac{2 π}{5}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{5} |}$

Step 3.3

$\frac{2 π}{5}$ is approximately $1.25663706$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{5}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{2 π}$

Step 3.5

Cancel the common factor of $2 π$ .

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Step 3.5.1

Cancel the common factor.

$2 π \frac{5}{2 π}$

Step 3.5.2

Rewrite the expression.

$5$

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: $\frac{\frac{5}{4}}{\frac{2 π}{5}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{5}{4} \cdot \frac{5}{2 π}$

Step 4.4

Multiply $\frac{5}{4} \cdot \frac{5}{2 π}$ .

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Step 4.4.1

Multiply $\frac{5}{4}$ by $\frac{5}{2 π}$ .

Phase Shift: $\frac{5 \cdot 5}{4 (2 π)}$

Step 4.4.2

Multiply $5$ by $5$ .

Phase Shift: $\frac{25}{4 (2 π)}$

Step 4.4.3

Multiply $2$ by $4$ .

Phase Shift: $\frac{25}{8 π}$

Step 5

List the properties of the trigonometric function.

Amplitude: $4$

Period: $5$

Phase Shift: $\frac{25}{8 π}$ ( $\frac{25}{8 π}$ to the right)

Vertical Shift: None

Step 6