Trigonometry Examples

$f (t) = - 6 \sin (\frac{3}{2} x + 4 π)$

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.

$a = - 6$

$b = \frac{3}{2}$

$c = - 4 π$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $6$

Step 3

Find the period of $- 6 \sin (\frac{3 x}{2} + 4 π)$ .

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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 $\frac{3}{2}$ in the formula for period.

$\frac{2 π}{| \frac{3}{2} |}$

Step 3.3

$\frac{3}{2}$ is approximately $1.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{3}{2}}$

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{2}{3}$

Step 3.5

Multiply $2 π \frac{2}{3}$ .

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Step 3.5.1

Combine $\frac{2}{3}$ and $2$ .

$\frac{2 \cdot 2}{3} π$

Step 3.5.2

Multiply $2$ by $2$ .

$\frac{4}{3} π$

Step 3.5.3

Combine $\frac{4}{3}$ and $π$ .

$\frac{4 π}{3}$

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: $\frac{- 4 π}{\frac{3}{2}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- 4 π \frac{2}{3}$

Step 4.4

Multiply $- 4 π \frac{2}{3}$ .

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Step 4.4.1

Combine $\frac{2}{3}$ and $- 4$ .

Phase Shift: $\frac{2 \cdot - 4}{3} \cdot π$

Step 4.4.2

Multiply $2$ by $- 4$ .

Phase Shift: $\frac{- 8}{3} \cdot π$

Step 4.4.3

Combine $\frac{- 8}{3}$ and $π$ .

Phase Shift: $\frac{- 8 π}{3}$

Step 4.5

Move the negative in front of the fraction.

Phase Shift: $- \frac{8 π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude: $6$

Period: $\frac{4 π}{3}$

Phase Shift: $- \frac{8 π}{3}$ ( $\frac{8 π}{3}$ to the left)

Vertical Shift: None

Step 6