Trigonometry Examples

y = \sin (3 x + 2 π)

$y = \sin (3 x + 2 π)$

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 = 3

$b = 3$

c = - 2 π

$c = - 2 π$

d = 0

$d = 0$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

1

$1$

Step 3

Find the period of

\sin (3 x + 2 π)

$\sin (3 x + 2 π)$ .

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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

3

$3$ in the formula for period.

\frac{2 π}{| 3 |}

$\frac{2 π}{| 3 |}$

Step 3.3

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

0

$0$ and

3

$3$ is

3

$3$ .

\frac{2 π}{3}

$\frac{2 π}{3}$

\frac{2 π}{3}

$\frac{2 π}{3}$

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{- 2 π}{3}

$\frac{- 2 π}{3}$

Step 4.3

Move the negative in front of the fraction.

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:

1

$1$

Period:

\frac{2 π}{3}

$\frac{2 π}{3}$

Phase Shift:

- \frac{2 π}{3}

$- \frac{2 π}{3}$ (

\frac{2 π}{3}

$\frac{2 π}{3}$ to the left)

Vertical Shift: None

Step 6