Trigonometry Examples

$y = \sin (π + 6 x)$

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.

$a = 1$

$b = 6$

$c = - π$

$d = 0$

Step 2

Find the amplitude

$| a |$ .

Amplitude:

$1$

Step 3

Find the period of

$\sin (π + 6 x)$ .

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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

$6$ in the formula for period.

$\frac{2 π}{| 6 |}$

Step 3.3

The absolute value is the distance between a number and zero. The distance between

$0$ and

$6$ is

$6$ .

$\frac{2 π}{6}$

Step 3.4

Cancel the common factor of

$2$ and

$6$ .

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Step 3.4.1

Factor

$2$ out of

$2 π$ .

$\frac{2 (π)}{6}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor

$2$ out of

$6$ .

$\frac{2 π}{2 \cdot 3}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 π}{2 \cdot 3}$

Step 3.4.2.3

Rewrite the expression.

$\frac{π}{3}$

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:

$\frac{- π}{6}$

Step 4.3

Move the negative in front of the fraction.

Phase Shift:

$- \frac{π}{6}$

Phase Shift:

$- \frac{π}{6}$

Step 5

List the properties of the trigonometric function.

Amplitude:

$1$

Period:

$\frac{π}{3}$

Phase Shift:

$- \frac{π}{6}$ (

$\frac{π}{6}$ to the left)

Vertical Shift: None

Step 6