Find Amplitude, Period, and Phase Shift y=1/2sin(1/2x+pi/2)

$y = \frac{1}{2} \sin (\frac{1}{2} x + \frac{π}{2})$

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

$a = \frac{1}{2}$

$b = \frac{1}{2}$

$c = - \frac{π}{2}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $\frac{1}{2}$

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

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{1}{2}$ in the formula for period.

$\frac{2 π}{| \frac{1}{2} |}$

$\frac{1}{2}$ is approximately $0.5$ which is positive so remove the absolute value

$\frac{2 π}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \cdot 2$

Multiply $2$ by $2$ .

$4 π$

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: $\frac{- \frac{π}{2}}{\frac{1}{2}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{2} \cdot 2$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{π}{2}$ into the numerator.

Phase Shift: $\frac{- π}{2} \cdot 2$

Cancel the common factor.

Phase Shift: $\frac{- π}{2} \cdot 2$

Rewrite the expression.

Phase Shift: $- π$

List the properties of the trigonometric function.

Amplitude: $\frac{1}{2}$

Period: $4 π$

Phase Shift: $- π$ ( $π$ to the left)

Vertical Shift: None

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Name

Name	seven hundred sixty-seven million three hundred ninety-four thousand seven hundred sixty-one

Interesting facts

767394761 has 16 divisors, whose sum is 890529024
The reverse of 767394761 is 167493767
Previous prime number is 967

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 50
Digital Root 5

Name

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

Interesting facts

1886980695 has 32 divisors, whose sum is 3119800320
The reverse of 1886980695 is 5960896881
Previous prime number is 2729

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 60
Digital Root 6

Name

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

Interesting facts

1263450042 has 16 divisors, whose sum is 2807666880
The reverse of 1263450042 is 2400543621
Previous prime number is 9

Basic properties

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

Graph csc(x+pi/3)

Simplify sin(-300)

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