Trigonometry Examples

$y = \sin (2 π x)$

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.

$a = 1$

$b = 2 π$

$c = 0$

$d = 0$

Step 2

Find the amplitude

$| a |$ .

Amplitude:

$1$

Step 3

Find the period of

$\sin (2 π x)$ .

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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

$2 π$ in the formula for period.

$\frac{2 π}{| 2 π |}$

Step 3.3

$2 π$ is approximately

$6.2831853$ which is positive so remove the absolute value

$\frac{2 π}{2 π}$

Step 3.4

Cancel the common factor of

$2$ .

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Step 3.4.1

Cancel the common factor.

$\frac{2 π}{2 π}$

Step 3.4.2

Rewrite the expression.

$\frac{π}{π}$

Step 3.5

Cancel the common factor of

$π$ .

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Step 3.5.1

Cancel the common factor.

$\frac{π}{π}$

Step 3.5.2

Rewrite the expression.

$1$

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:

$\frac{0}{2 π}$

Step 4.3

Cancel the common factor of

$0$ and

$2$ .

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Step 4.3.1

Factor

$2$ out of

$0$ .

Phase Shift:

$\frac{2 (0)}{2 π}$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor

$2$ out of

$2 π$ .

Phase Shift:

$\frac{2 (0)}{2 (π)}$

Step 4.3.2.2

Cancel the common factor.

Phase Shift:

$\frac{2 \cdot 0}{2 π}$

Step 4.3.2.3

Rewrite the expression.

Phase Shift:

$\frac{0}{π}$

Phase Shift:

$\frac{0}{π}$

Phase Shift:

$\frac{0}{π}$

Step 4.4

Divide

$0$ by

$π$ .

Phase Shift:

$0$

Phase Shift:

$0$

Step 5

List the properties of the trigonometric function.

Amplitude:

$1$

Period:

$1$

Phase Shift: None

Vertical Shift: None

Step 6