Find Amplitude, Period, and Phase Shift y=5sin((2x)/3-pi)

$y = 5 \sin (\frac{2 x}{3} - π)$

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

$a = 5$

$b = \frac{2}{3}$

$c = π$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $5$

Find the period using the formula $\frac{2 π}{| b |}$ .

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

Period: $\frac{2 π}{| b |}$

Replace $b$ with $\frac{2}{3}$ in the formula for period.

Period: $\frac{2 π}{| \frac{2}{3} |}$

Solve the equation.

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$\frac{2}{3}$ is approximately $0.\overline{6}$ which is positive so remove the absolute value

Period: $\frac{2 π}{\frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Period: $2 π (\frac{3}{2})$

Cancel the common factor of $2$ .

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Factor $2$ out of $2 π$ .

Period: $2 (π) (\frac{3}{2})$

Cancel the common factor.

Period: $2 π (\frac{3}{2})$

Rewrite the expression.

Period: $π \cdot 3$

Move $3$ to the left of $π$ .

Period: $3 π$

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: $\frac{π}{\frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $π (\frac{3}{2})$

Combine $π$ and $\frac{3}{2}$ .

Phase Shift: $\frac{π \cdot 3}{2}$

Move $3$ to the left of $π$ .

Phase Shift: $\frac{3 π}{2}$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: $5$

Period: $3 π$

Phase Shift: $\frac{3 π}{2}$ ( $\frac{3 π}{2}$ to the right)

Vertical Shift: $0$

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Name

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

Interesting facts

1399569404 has 16 divisors, whose sum is 3334268448
The reverse of 1399569404 is 4049659931
Previous prime number is 17

Basic properties

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

Name

Name	one billion eight hundred fifty-eight million thirty-two thousand thirty-eight

Interesting facts

1858032038 has 8 divisors, whose sum is 2814107088
The reverse of 1858032038 is 8302308581
Previous prime number is 103

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 38
Digital Root 2

Name

Name	one billion eight hundred ninety million six hundred ninety-two thousand two hundred sixty-nine

Interesting facts

1890692269 has 8 divisors, whose sum is 1902391160
The reverse of 1890692269 is 9622960981
Previous prime number is 193

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 52
Digital Root 7

Find the Exact Value cos(46)

Rationalize the Denominator (1-cos(x))/(1+cos(x))*(sin(x))/(sin(x))

Convert from Degrees to Radians 68

Find the Inverse 1/(csc(x)^2)+1/(sec(x)^2)

Factor sin(theta)