Find Amplitude, Period, and Phase Shift f(t)=cos(5t+pi/2)-3

$f (t) = \cos (5 t + \frac{π}{2}) - 3$

Step 1

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

$a = 1$

$b = 5$

$c = - \frac{π}{2}$

$d = - 3$

Step 2

Find the amplitude $| a |$ .

Amplitude: $1$

Step 3

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

Find the period of $\cos (5 t + \frac{π}{2})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $5$ in the formula for period.

$\frac{2 π}{| 5 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{2 π}{5}$

Find the period of $- 3$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $5$ in the formula for period.

$\frac{2 π}{| 5 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{2 π}{5}$

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

$\frac{2 π}{5}$

Step 4

Find the phase shift using the formula $\frac{c}{b}$ .

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{- \frac{π}{2}}{5}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{2} \cdot \frac{1}{5}$

Multiply $- \frac{π}{2} \cdot \frac{1}{5}$ .

Multiply $\frac{1}{5}$ by $\frac{π}{2}$ .

Phase Shift: $- \frac{π}{5 \cdot 2}$

Multiply $5$ by $2$ .

Phase Shift: $- \frac{π}{10}$

Step 5

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{2 π}{5}$

Phase Shift: $- \frac{π}{10}$ ( $\frac{π}{10}$ to the left)

Vertical Shift: $- 3$

Step 6

Find Amplitude, Period, and Phase Shift f(t)=cos(5t+pi/2)-3

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