Trigonometry Examples

y = \frac{1}{4} sin (8 π x - \frac{5 π}{4})

$y = \frac{1}{4} \sin (8 π x - \frac{5 π}{4})$

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 = 8 π

$b = 8 π$

c = \frac{5 π}{4}

$c = \frac{5 π}{4}$

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 (8 π x - \frac{5 π}{4})}{4}

$\frac{\sin (8 π x - \frac{5 π}{4})}{4}$ .

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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

8 π

$8 π$ in the formula for period.

\frac{2 π}{| 8 π |}

$\frac{2 π}{| 8 π |}$

Step 3.3

8 π

$8 π$ is approximately

25.13274122

$25.13274122$ which is positive so remove the absolute value

\frac{2 π}{8 π}

$\frac{2 π}{8 π}$

Step 3.4

Cancel the common factor of

2

$2$ and

8

$8$ .

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Step 3.4.1

Factor

2

$2$ out of

2 π

$2 π$ .

\frac{2 (π)}{8 π}

$\frac{2 (π)}{8 π}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor

2

$2$ out of

8 π

$8 π$ .

\frac{2 (π)}{2 (4 π)}

$\frac{2 (π)}{2 (4 π)}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 π}{2 (4 π)}$

Step 3.4.2.3

Rewrite the expression.

$\frac{π}{4 π}$

Step 3.5

Cancel the common factor of

$π$ .

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Step 3.5.1

Cancel the common factor.

$\frac{π}{4 π}$

Step 3.5.2

Rewrite the expression.

$\frac{1}{4}$

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:

$\frac{\frac{5 π}{4}}{8 π}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift:

$\frac{5 π}{4} \cdot \frac{1}{8 π}$

Step 4.4

Cancel the common factor of

$π$ .

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Step 4.4.1

Factor

$π$ out of

$5 π$ .

Phase Shift:

$\frac{π \cdot 5}{4} \cdot \frac{1}{8 π}$

Step 4.4.2

Factor

$π$ out of

$8 π$ .

Phase Shift:

$\frac{π \cdot 5}{4} \cdot \frac{1}{π \cdot 8}$

Step 4.4.3

Cancel the common factor.

Phase Shift:

$\frac{π \cdot 5}{4} \cdot \frac{1}{π \cdot 8}$

Step 4.4.4

Rewrite the expression.

Phase Shift:

$\frac{5}{4} \cdot \frac{1}{8}$

Phase Shift:

$\frac{5}{4} \cdot \frac{1}{8}$

Step 4.5

Multiply

$\frac{5}{4}$ by

$\frac{1}{8}$ .

Phase Shift:

$\frac{5}{4 \cdot 8}$

Step 4.6

Multiply

$4$ by

$8$ .

Phase Shift:

$\frac{5}{32}$

Phase Shift:

$\frac{5}{32}$

Step 5

List the properties of the trigonometric function.

Amplitude:

$\frac{1}{4}$

Period:

$\frac{1}{4}$

Phase Shift:

$\frac{5}{32}$ (

$\frac{5}{32}$ to the right)

Vertical Shift: None

Step 6