Trigonometry Examples

$y = - 6 \sin (- 4 x + \frac{2}{π})$

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.

$a = - 6$

$b = - 4$

$c = - \frac{2}{π}$

$d = 0$

Step 2

Find the amplitude $| a |$ .

Amplitude: $6$

Step 3

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

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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 $- 4$ in the formula for period.

$\frac{2 π}{| - 4 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between $- 4$ and $0$ is $4$ .

$\frac{2 π}{4}$

Step 3.4

Cancel the common factor of $2$ and $4$ .

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Step 3.4.1

Factor $2$ out of $2 π$ .

$\frac{2 (π)}{4}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $2$ out of $4$ .

$\frac{2 π}{2 \cdot 2}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 2}$

Step 3.4.2.3

Rewrite the expression.

$\frac{π}{2}$

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: $\frac{- \frac{2}{π}}{- 4}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{2}{π} \cdot \frac{1}{- 4}$

Step 4.4

Cancel the common factor of $2$ .

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Step 4.4.1

Move the leading negative in $- \frac{2}{π}$ into the numerator.

Phase Shift: $\frac{- 2}{π} \cdot \frac{1}{- 4}$

Step 4.4.2

Factor $2$ out of $- 2$ .

Phase Shift: $\frac{2 (- 1)}{π} \cdot \frac{1}{- 4}$

Step 4.4.3

Factor $2$ out of $- 4$ .

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

Step 4.4.4

Cancel the common factor.

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

Step 4.4.5

Rewrite the expression.

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

Step 4.5

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

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

Step 4.6

Move $- 2$ to the left of $π$ .

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

Step 4.7

Dividing two negative values results in a positive value.

Phase Shift: $\frac{1}{2 π}$

Step 5

List the properties of the trigonometric function.

Amplitude: $6$

Period: $\frac{π}{2}$

Phase Shift: $\frac{1}{2 π}$ ( $\frac{1}{2 π}$ to the right)

Vertical Shift: None

Step 6