Trigonometry Examples

y = \sin (8 x)

$y = \sin (8 x)$

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 = 1

$a = 1$

b = 8

$b = 8$

c = 0

$c = 0$

d = 0

$d = 0$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

1

$1$

Step 3

Find the period of

\sin (8 x)

$\sin (8 x)$ .

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Step 3.1

The period of the function can be calculated using

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$

Step 3.2

Replace

b

$b$ with

8

$8$ in the formula for period.

\frac{2 π}{| 8 |}

$\frac{2 π}{| 8 |}$

Step 3.3

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

0

$0$ and

8

$8$ is

8

$8$ .

\frac{2 π}{8}

$\frac{2 π}{8}$

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}

$\frac{2 (π)}{8}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor

2

$2$ out of

8

$8$ .

\frac{2 π}{2 \cdot 4}

$\frac{2 π}{2 \cdot 4}$

Step 3.4.2.2

Cancel the common factor.

\frac{2 π}{2 \cdot 4}

$\frac{2 π}{2 \cdot 4}$

Step 3.4.2.3

Rewrite the expression.

\frac{π}{4}

$\frac{π}{4}$

\frac{π}{4}

$\frac{π}{4}$

\frac{π}{4}

$\frac{π}{4}$

\frac{π}{4}

$\frac{π}{4}$

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{0}{8}

$\frac{0}{8}$

Step 4.3

Divide

0

$0$ by

8

$8$ .

Phase Shift:

0

$0$

Phase Shift:

0

$0$

Step 5

List the properties of the trigonometric function.

Amplitude:

1

$1$

Period:

\frac{π}{4}

$\frac{π}{4}$

Phase Shift: None

Vertical Shift: None

Step 6