Trigonometry Examples

$y = \sin (3 θ)$

Step 1

Use the form

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

$a = 1$

$b = 3$

$c = 0$

$d = 0$

Step 2

Find the amplitude

$| a |$ .

Amplitude:

$1$

Step 3

Find the period of

$\sin (3 θ)$ .

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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

$3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 3.3

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

$0$ and

$3$ is

$3$ .

$\frac{2 π}{3}$

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:

$\frac{0}{3}$

Step 4.3

Divide

$0$ by

$3$ .

Phase Shift:

$0$

Phase Shift:

$0$

Step 5

List the properties of the trigonometric function.

Amplitude:

$1$

Period:

$\frac{2 π}{3}$

Phase Shift: None

Vertical Shift: None

Step 6