Find Amplitude, Period, and Phase Shift y=cos((6pix)/7+pi/2)

$y = \cos (\frac{6 π x}{7} + \frac{π}{2})$

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

$a = 1$

$b = \frac{6 π}{7}$

$c = - \frac{π}{2}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $1$

Find the period of $\cos (\frac{6 π x}{7} + \frac{π}{2})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{6 π}{7}$ in the formula for period.

$\frac{2 π}{| \frac{6 π}{7} |}$

$\frac{6 π}{7}$ is approximately $2.6927937$ which is positive so remove the absolute value

$\frac{2 π}{\frac{6 π}{7}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{7}{6 π}$

Cancel the common factor of $2 π$ .

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Factor $2 π$ out of $6 π$ .

$2 π \frac{7}{2 π (3)}$

Cancel the common factor.

$2 π \frac{7}{2 π \cdot 3}$

Rewrite the expression.

$\frac{7}{3}$

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: $\frac{- \frac{π}{2}}{\frac{6 π}{7}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{2} \cdot \frac{7}{6 π}$

Cancel the common factor of $π$ .

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Move the leading negative in $- \frac{π}{2}$ into the numerator.

Phase Shift: $\frac{- π}{2} \cdot \frac{7}{6 π}$

Factor $π$ out of $- π$ .

Phase Shift: $\frac{π \cdot - 1}{2} \cdot \frac{7}{6 π}$

Factor $π$ out of $6 π$ .

Phase Shift: $\frac{π \cdot - 1}{2} \cdot \frac{7}{π \cdot 6}$

Cancel the common factor.

Phase Shift: $\frac{π \cdot - 1}{2} \cdot \frac{7}{π \cdot 6}$

Rewrite the expression.

Phase Shift: $\frac{- 1}{2} \cdot \frac{7}{6}$

Multiply $\frac{- 1}{2}$ and $\frac{7}{6}$ .

Phase Shift: $\frac{- 1 \cdot 7}{2 \cdot 6}$

Simplify the expression.

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Multiply $- 1$ by $7$ .

Phase Shift: $\frac{- 7}{2 \cdot 6}$

Multiply $2$ by $6$ .

Phase Shift: $\frac{- 7}{12}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{7}{12}$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{7}{3}$

Phase Shift: $- \frac{7}{12}$ ( $\frac{7}{12}$ to the left)

Vertical Shift: None

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Name

Name	one billion nine hundred eighty-two million two hundred thirteen thousand seven hundred sixty-nine

Interesting facts

1982213769 has 4 divisors, whose sum is 2642951696
The reverse of 1982213769 is 9673122891
Previous prime number is 3

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 48
Digital Root 3

Name

Name	one billion five hundred ninety-three million six hundred thirty-two thousand four hundred twenty-eight

Interesting facts

1593632428 has 32 divisors, whose sum is 3800251728
The reverse of 1593632428 is 8242363951
Previous prime number is 17

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 43
Digital Root 7

Name

Name	two billion seventy-four million nine hundred forty-eight thousand four hundred thirty-nine

Interesting facts

2074948439 has 8 divisors, whose sum is 2084040000
The reverse of 2074948439 is 9348494702
Previous prime number is 719

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 50
Digital Root 5

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