Find Amplitude, Period, and Phase Shift y=sin(1/3(x-pi/6))

$y = \sin (\frac{1}{3} (x - \frac{π}{6}))$

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

$a = 1$

$b = \frac{1}{3}$

$c = \frac{π}{18}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $1$

Find the period of $\sin (\frac{x}{3} - \frac{π}{18})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{3}$ in the formula for period.

$\frac{2 π}{| \frac{1}{3} |}$

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 3$

Multiply $3$ by $2$ .

$6 π$

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: $\frac{\frac{π}{18}}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{18} \cdot 3$

Cancel the common factor of $3$ .

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Factor $3$ out of $18$ .

Phase Shift: $\frac{π}{3 (6)} \cdot 3$

Cancel the common factor.

Phase Shift: $\frac{π}{3 \cdot 6} \cdot 3$

Rewrite the expression.

Phase Shift: $\frac{π}{6}$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $6 π$

Phase Shift: $\frac{π}{6}$ ( $\frac{π}{6}$ to the right)

Vertical Shift: None

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Name

Name	one billion three hundred seventy million six hundred sixty-two thousand four hundred forty-one

Interesting facts

1370662441 has 4 divisors, whose sum is 1371917572
The reverse of 1370662441 is 1442660731
Previous prime number is 1093

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 34
Digital Root 7

Name

Name	nine hundred ninety-one million ninety-eight thousand two hundred fifty-five

Interesting facts

991098255 has 8 divisors, whose sum is 994165248
The reverse of 991098255 is 552890199
Previous prime number is 1051

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 48
Digital Root 3

Name

Name	four hundred ninety-five million eight hundred eighty-nine thousand four hundred fifty-eight

Interesting facts

495889458 has 16 divisors, whose sum is 994592736
The reverse of 495889458 is 854988594
Previous prime number is 353

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 60
Digital Root 6

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