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

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

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

$a = 4$

$b = \frac{2 π}{5}$

$c = \frac{5}{4}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $4$

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{2 π}{5}$ in the formula for period.

$\frac{2 π}{| \frac{2 π}{5} |}$

$\frac{2 π}{5}$ is approximately $1.25663706$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2 π}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{5}{2 π}$

Cancel the common factor of $2 π$ .

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Cancel the common factor.

$2 π \frac{5}{2 π}$

Rewrite the expression.

$5$

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: $\frac{\frac{5}{4}}{\frac{2 π}{5}}$

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{5}{4} \cdot \frac{5}{2 π}$ .

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

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

Multiply $5$ by $5$ .

Phase Shift: $\frac{25}{4 (2 π)}$

Multiply $2$ by $4$ .

Phase Shift: $\frac{25}{8 π}$

List the properties of the trigonometric function.

Amplitude: $4$

Period: $5$

Phase Shift: $\frac{25}{8 π}$ ( $\frac{25}{8 π}$ to the right)

Vertical Shift: None

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Name

Name	one billion eight hundred seventy-eight million thirty thousand four hundred eighty-three

Interesting facts

1878030483 has 8 divisors, whose sum is 1987745760
The reverse of 1878030483 is 3840308781
Previous prime number is 97

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 42
Digital Root 6

Name

Name	two billion ninety million six hundred twenty-nine thousand seventy-eight

Interesting facts

2090629078 has 8 divisors, whose sum is 3149065440
The reverse of 2090629078 is 8709260902
Previous prime number is 239

Basic properties

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

Name

Name	one billion eight hundred seven million one hundred eighty-six thousand nine hundred sixty-nine

Interesting facts

1807186969 has 8 divisors, whose sum is 2069324928
The reverse of 1807186969 is 9696817081
Previous prime number is 521

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 55
Digital Root 1

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

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Verify the Identity (cos(x))/(sec(x)-1)-(cos(x))/(sec(x)+1)=(2cos(x))/(tan(x)^2)