Find Amplitude, Period, and Phase Shift y=3csc(theta/4+5)

$y = 3 \csc (\frac{θ}{4} + 5)$

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

$a = 3$

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

$c = - 5$

$d = 0$

Since the graph of the function $c s c$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Find the period of $3 \csc (\frac{θ}{4} + 5)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{4}$ in the formula for period.

$\frac{2 π}{| \frac{1}{4} |}$

$\frac{1}{4}$ is approximately $0.25$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 4$

Multiply $4$ by $2$ .

$8 π$

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: $\frac{- 5}{\frac{1}{4}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- 5 \cdot 4$

Multiply $- 5$ by $4$ .

Phase Shift: $- 20$

List the properties of the trigonometric function.

Amplitude: None

Period: $8 π$

Phase Shift: $- 20$ ( $20$ to the left)

Vertical Shift: None

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Name

Name	one billion twenty-nine million one hundred forty-four thousand five hundred one

Interesting facts

1029144501 has 16 divisors, whose sum is 1235584896
The reverse of 1029144501 is 1054419201
Previous prime number is 27

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 27
Digital Root 9

Name

Name	one billion three hundred thirty-seven million nine hundred fifty-one thousand fifty-two

Interesting facts

1337951052 has 64 divisors, whose sum is 5368947840
The reverse of 1337951052 is 2501597331
Previous prime number is 313

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 36
Digital Root 9

Name

Name	one billion four hundred million nine hundred eighty-six thousand six hundred ninety-five

Interesting facts

1400986695 has 8 divisors, whose sum is 1495473408
The reverse of 1400986695 is 5966890041
Previous prime number is 1403

Basic properties

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

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