Find Amplitude, Period, and Phase Shift y=1/2cos((4pix)/5-pi/2)

$y = \frac{1}{2} \cos (\frac{4 π x}{5} - \frac{π}{2})$

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

$a = \frac{1}{2}$

$b = \frac{4 π}{5}$

$c = \frac{π}{2}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{2}$

Find the period of $\frac{\cos (\frac{4 π x}{5} - \frac{π}{2})}{2}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{4 π}{5}$ in the formula for period.

$\frac{2 π}{| \frac{4 π}{5} |}$

$\frac{4 π}{5}$ is approximately $2.51327412$ which is positive so remove the absolute value

$\frac{2 π}{\frac{4 π}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{4 π}$

Cancel the common factor of $2 π$ .

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Factor $2 π$ out of $4 π$ .

$2 π \frac{5}{2 π (2)}$

Cancel the common factor.

$2 π \frac{5}{2 π \cdot 2}$

Rewrite the expression.

$\frac{5}{2}$

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

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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: $\frac{\frac{π}{2}}{\frac{4 π}{5}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{2} \cdot \frac{5}{4 π}$

Cancel the common factor of $π$ .

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Factor $π$ out of $4 π$ .

Phase Shift: $\frac{π}{2} \cdot \frac{5}{π \cdot 4}$

Cancel the common factor.

Phase Shift: $\frac{π}{2} \cdot \frac{5}{π \cdot 4}$

Rewrite the expression.

Phase Shift: $\frac{1}{2} \cdot \frac{5}{4}$

Multiply $\frac{1}{2}$ and $\frac{5}{4}$ .

Phase Shift: $\frac{5}{2 \cdot 4}$

Multiply $2$ by $4$ .

Phase Shift: $\frac{5}{8}$

List the properties of the trigonometric function.

Amplitude: $\frac{1}{2}$

Period: $\frac{5}{2}$

Phase Shift: $\frac{5}{8}$ ( $\frac{5}{8}$ to the right)

Vertical Shift: None

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Name

Name	eight hundred twenty-five million three hundred sixty thousand three hundred twenty-six

Interesting facts

825360326 has 16 divisors, whose sum is 1246687200
The reverse of 825360326 is 623063528
Previous prime number is 269

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 35
Digital Root 8

Name

Name	one billion six hundred fifty-seven million nine hundred nine thousand seven hundred ninety-four

Interesting facts

1657909794 has 8 divisors, whose sum is 2487134160
The reverse of 1657909794 is 4979097561
Previous prime number is 10443

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 57
Digital Root 3

Name

Name	five hundred sixty-four million two hundred fifty-nine thousand six hundred fifty-five

Interesting facts

564259655 has 8 divisors, whose sum is 678493584
The reverse of 564259655 is 556952465
Previous prime number is 491

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 47
Digital Root 2

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