Find Amplitude, Period, and Phase Shift y=1/4cos((2x)/7+1/3)

$y = \frac{1}{4} \cos (\frac{2 x}{7} + \frac{1}{3})$

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

$a = \frac{1}{4}$

$b = \frac{2}{7}$

$c = - \frac{1}{3}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{4}$

Find the period of $\frac{\cos (\frac{2 x}{7} + \frac{1}{3})}{4}$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{2}{7}$ in the formula for period.

$\frac{2 π}{| \frac{2}{7} |}$

$\frac{2}{7}$ is approximately $0.\overline{285714}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{7}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{7}{2}$

Cancel the common factor of $2$ .

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Factor $2$ out of $2 π$ .

$2 (π) \frac{7}{2}$

Cancel the common factor.

$2 π \frac{7}{2}$

Rewrite the expression.

$π \cdot 7$

Move $7$ to the left of $π$ .

$7 π$

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: $\frac{- \frac{1}{3}}{\frac{2}{7}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{1}{3} \cdot \frac{7}{2}$

Multiply $- \frac{1}{3} \cdot \frac{7}{2}$ .

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Multiply $\frac{7}{2}$ and $\frac{1}{3}$ .

Phase Shift: $- \frac{7}{2 \cdot 3}$

Multiply $2$ by $3$ .

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

List the properties of the trigonometric function.

Amplitude: $\frac{1}{4}$

Period: $7 π$

Phase Shift: $- \frac{7}{6}$ ( $\frac{7}{6}$ to the left)

Vertical Shift: None

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Name

Name	one hundred fifty-seven million six hundred five thousand four hundred fifty-two

Interesting facts

157605452 has 8 divisors, whose sum is 354612276
The reverse of 157605452 is 254506751
Previous prime number is 2

Basic properties

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

Name

Name	one billion one hundred sixty-nine million four hundred eighteen thousand three hundred fifty-five

Interesting facts

1169418355 has 32 divisors, whose sum is 1713239424
The reverse of 1169418355 is 5538149611
Previous prime number is 113

Basic properties

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

Name

Name	one billion seven hundred thirty-four million eight hundred fifty-nine thousand one hundred sixty-three

Interesting facts

1734859163 has 8 divisors, whose sum is 1933589520
The reverse of 1734859163 is 3619584371
Previous prime number is 19

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 47
Digital Root 2

Graph v=11i+11j

Evaluate -5/(2 square root of 6)

Convert from Degrees to Radians 40

Simplify cos(x)^2csc(x)-csc(x)

Find the Domain v=-2i-2j