Precalculus Examples

$P (t) = 20 \sin (\frac{12 π}{5} t) + 120$

Step 1

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

$a = 20$

$b = \frac{12 π}{5}$

$c = 0$

$d = 120$

Step 2

Find the amplitude $| a |$ .

Amplitude: $20$

Step 3

Find the period using the formula $\frac{2 π}{| b |}$ .

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Step 3.1

Find the period of $20 \sin (\frac{12 π t}{5})$ .

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Step 3.1.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.1.2

Replace $b$ with $\frac{12 π}{5}$ in the formula for period.

$\frac{2 π}{| \frac{12 π}{5} |}$

Step 3.1.3

$\frac{12 π}{5}$ is approximately $7.53982236$ which is positive so remove the absolute value

$\frac{2 π}{\frac{12 π}{5}}$

Step 3.1.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{12 π}$

Step 3.1.5

Cancel the common factor of $2 π$ .

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Step 3.1.5.1

Factor $2 π$ out of $12 π$ .

$2 π \frac{5}{2 π (6)}$

Step 3.1.5.2

Cancel the common factor.

$2 π \frac{5}{2 π \cdot 6}$

Step 3.1.5.3

Rewrite the expression.

$\frac{5}{6}$

Step 3.2

Find the period of $120$ .

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Step 3.2.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2.2

Replace $b$ with $\frac{12 π}{5}$ in the formula for period.

$\frac{2 π}{| \frac{12 π}{5} |}$

Step 3.2.3

$\frac{12 π}{5}$ is approximately $7.53982236$ which is positive so remove the absolute value

$\frac{2 π}{\frac{12 π}{5}}$

Step 3.2.4

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{12 π}$

Step 3.2.5

Cancel the common factor of $2 π$ .

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Step 3.2.5.1

Factor $2 π$ out of $12 π$ .

$2 π \frac{5}{2 π (6)}$

Step 3.2.5.2

Cancel the common factor.

$2 π \frac{5}{2 π \cdot 6}$

Step 3.2.5.3

Rewrite the expression.

$\frac{5}{6}$

Step 3.3

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

$\frac{5}{6}$

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: $\frac{0}{\frac{12 π}{5}}$

Step 4.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 (\frac{5}{12 π})$

Step 4.4

Multiply $0$ by $\frac{5}{12 π}$ .

Phase Shift: $0$

Step 5

List the properties of the trigonometric function.

Amplitude: $20$

Period: $\frac{5}{6}$

Phase Shift: None

Vertical Shift: $120$

Step 6