Trigonometry Examples

f (t) = 0.2 sin (t - 0.3) + 0.1

$f (t) = 0.2 \sin (t - 0.3) + 0.1$

Step 1

Use the form

a sin (b t - c) + d

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

a = 0.2

$a = 0.2$

b = 1

$b = 1$

c = 0.3

$c = 0.3$

d = 0.1

$d = 0.1$

Step 2

Find the amplitude

| a |

$| a |$ .

Amplitude:

0.2

$0.2$

Step 3

Find the period using the formula

\frac{2 π}{| b |}

$\frac{2 π}{| b |}$ .

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Step 3.1

Find the period of

0.2 sin (t - 0.3)

$0.2 \sin (t - 0.3)$ .

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Step 3.1.1

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 3.1.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 3.1.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 3.1.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 3.2

Find the period of

0.1

$0.1$ .

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Step 3.2.1

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 3.2.2

Replace

b

$b$ with

1

$1$ in the formula for period.

\frac{2 π}{| 1 |}

$\frac{2 π}{| 1 |}$

Step 3.2.3

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

0

$0$ and

1

$1$ is

1

$1$ .

\frac{2 π}{1}

$\frac{2 π}{1}$

Step 3.2.4

Divide

2 π

$2 π$ by

1

$1$ .

2 π

$2 π$

2 π

$2 π$

Step 3.3

The period of addition/subtraction of trig functions is the maximum of the individual periods.

2 π

$2 π$

2 π

$2 π$

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{0.3}{1}

$\frac{0.3}{1}$

Step 4.3

Divide

0.3

$0.3$ by

1

$1$ .

Phase Shift:

0.3

$0.3$

Phase Shift:

0.3

$0.3$

Step 5

List the properties of the trigonometric function.

Amplitude:

0.2

$0.2$

Period:

2 π

$2 π$

Phase Shift:

0.3

$0.3$ (

0.3

$0.3$ to the right)

Vertical Shift:

0.1

$0.1$

Step 6