Trigonometry Examples

y = \frac{1}{4} sin (x - \frac{2 π}{3})

$y = \frac{1}{4} \sin (x - \frac{2 π}{3})$

Step 1

Use the form

a sin (b x - c) + d

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

a = \frac{1}{4}

$a = \frac{1}{4}$

b = 1

$b = 1$

c = \frac{2 π}{3}

$c = \frac{2 π}{3}$

d = 0

$d = 0$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

\frac{1}{4}

$\frac{1}{4}$

Step 3

Find the period of

\frac{sin (x - \frac{2 π}{3})}{4}

$\frac{\sin (x - \frac{2 π}{3})}{4}$ .

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Step 3.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 3.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 3.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 3.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 4

Find the phase shift using the formula

\frac{c}{b}

$\frac{c}{b}$ .

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Step 4.1

The phase shift of the function can be calculated from

\frac{c}{b}

$\frac{c}{b}$ .

Phase Shift:

\frac{c}{b}

$\frac{c}{b}$

Step 4.2

Replace the values of

c

$c$ and

b

$b$ in the equation for phase shift.

Phase Shift:

\frac{\frac{2 π}{3}}{1}

$\frac{\frac{2 π}{3}}{1}$

Step 4.3

Divide

\frac{2 π}{3}

$\frac{2 π}{3}$ by

1

$1$ .

Phase Shift:

\frac{2 π}{3}

$\frac{2 π}{3}$

Phase Shift:

\frac{2 π}{3}

$\frac{2 π}{3}$

Step 5

List the properties of the trigonometric function.

Amplitude:

\frac{1}{4}

$\frac{1}{4}$

Period:

2 π

$2 π$

Phase Shift:

\frac{2 π}{3}

$\frac{2 π}{3}$ (

\frac{2 π}{3}

$\frac{2 π}{3}$ to the right)

Vertical Shift: None

Step 6