Trigonometry Examples

$y = - 2 \sin (- 3 x + \frac{π}{2})$

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.

$a = - 2$

$b = - 3$

$c = - \frac{π}{2}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $2$

Step 3

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

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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 $- 3$ in the formula for period.

$\frac{2 π}{| - 3 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $- 3$ and $0$ is $3$ .

$\frac{2 π}{3}$

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: $\frac{- \frac{π}{2}}{- 3}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{2} \cdot \frac{1}{- 3}$

Step 4.4

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{2} \cdot (- \frac{1}{3})$

Step 4.5

Multiply $- \frac{π}{2} (- \frac{1}{3})$ .

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Step 4.5.1

Multiply $- 1$ by $- 1$ .

Phase Shift: $1 (\frac{π}{2}) \cdot \frac{1}{3}$

Step 4.5.2

Multiply $\frac{π}{2}$ by $1$ .

Phase Shift: $\frac{π}{2} \cdot \frac{1}{3}$

Step 4.5.3

Multiply $\frac{π}{2}$ by $\frac{1}{3}$ .

Phase Shift: $\frac{π}{2 \cdot 3}$

Step 4.5.4

Multiply $2$ by $3$ .

Phase Shift: $\frac{π}{6}$

Step 5

List the properties of the trigonometric function.

Amplitude: $2$

Period: $\frac{2 π}{3}$

Phase Shift: $\frac{π}{6}$ ( $\frac{π}{6}$ to the right)

Vertical Shift: None

Step 6