Graph y=sin(x+90)+2

$y = \sin (x + 90) + 2$

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

$a = 1$

$b = 1$

$c = - 90$

$d = 2$

Find the amplitude $| a |$ .

Amplitude: $1$

Find the period of $\sin (x + 90)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Divide $2 π$ by $1$ .

$2 π$

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: $\frac{- 90}{1}$

Divide $- 90$ by $1$ .

Phase Shift: $- 90$

Find the vertical shift $d$ .

Vertical Shift: $2$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $2 π$

Phase Shift: $- 90$ ( $90$ to the left)

Vertical Shift: $2$

Select a few points to graph.

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Find the point at $x = \frac{π}{2} - 90$ .

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Replace the variable $x$ with $\frac{π}{2} - 90$ in the expression.

$f (\frac{π}{2} - 90) = \sin ((\frac{π}{2} - 90) + 90) + 2$

Simplify the result.

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Simplify each term.

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Add $- 90$ and $90$ .

$f (\frac{π}{2} - 90) = \sin (\frac{π}{2} + 0) + 2$

Add $\frac{π}{2}$ and $0$ .

$f (\frac{π}{2} - 90) = \sin (\frac{π}{2}) + 2$

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{π}{2} - 90) = 1 + 2$

Add $1$ and $2$ .

$f (\frac{π}{2} - 90) = 3$

The final answer is $3$ .

$3$

Find the point at $x = π - 90$ .

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Replace the variable $x$ with $π - 90$ in the expression.

$f (π - 90) = \sin ((π - 90) + 90) + 2$

Simplify the result.

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Simplify each term.

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Add $- 90$ and $90$ .

$f (π - 90) = \sin (π + 0) + 2$

Add $π$ and $0$ .

$f (π - 90) = \sin (π) + 2$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$f (π - 90) = \sin (0) + 2$

The exact value of $\sin (0)$ is $0$ .

$f (π - 90) = 0 + 2$

Add $0$ and $2$ .

$f (π - 90) = 2$

The final answer is $2$ .

$2$

Find the point at $x = \frac{3 π}{2} - 90$ .

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Replace the variable $x$ with $\frac{3 π}{2} - 90$ in the expression.

$f (\frac{3 π}{2} - 90) = \sin ((\frac{3 π}{2} - 90) + 90) + 2$

Simplify the result.

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Simplify each term.

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Add $- 90$ and $90$ .

$f (\frac{3 π}{2} - 90) = \sin (\frac{3 π}{2} + 0) + 2$

Add $\frac{3 π}{2}$ and $0$ .

$f (\frac{3 π}{2} - 90) = \sin (\frac{3 π}{2}) + 2$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the fourth quadrant.

$f (\frac{3 π}{2} - 90) = - \sin (\frac{π}{2}) + 2$

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{3 π}{2} - 90) = - 1 \cdot 1 + 2$

Multiply $- 1$ by $1$ .

$f (\frac{3 π}{2} - 90) = - 1 + 2$

Add $- 1$ and $2$ .

$f (\frac{3 π}{2} - 90) = 1$

The final answer is $1$ .

$1$

Find the point at $x = 2 π - 90$ .

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Replace the variable $x$ with $2 π - 90$ in the expression.

$f (2 π - 90) = \sin ((2 π - 90) + 90) + 2$

Simplify the result.

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Simplify each term.

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Add $- 90$ and $90$ .

$f (2 π - 90) = \sin (2 π + 0) + 2$

Add $2 π$ and $0$ .

$f (2 π - 90) = \sin (2 π) + 2$

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (2 π - 90) = \sin (0) + 2$

The exact value of $\sin (0)$ is $0$ .

$f (2 π - 90) = 0 + 2$

Add $0$ and $2$ .

$f (2 π - 90) = 2$

The final answer is $2$ .

$2$

Find the point at $x = \frac{5 π}{2} - 90$ .

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Replace the variable $x$ with $\frac{5 π}{2} - 90$ in the expression.

$f (\frac{5 π}{2} - 90) = \sin ((\frac{5 π}{2} - 90) + 90) + 2$

Simplify the result.

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Simplify each term.

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Add $- 90$ and $90$ .

$f (\frac{5 π}{2} - 90) = \sin (\frac{5 π}{2} + 0) + 2$

Add $\frac{5 π}{2}$ and $0$ .

$f (\frac{5 π}{2} - 90) = \sin (\frac{5 π}{2}) + 2$

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (\frac{5 π}{2} - 90) = \sin (\frac{π}{2}) + 2$

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$f (\frac{5 π}{2} - 90) = 1 + 2$

Add $1$ and $2$ .

$f (\frac{5 π}{2} - 90) = 3$

The final answer is $3$ .

$3$

List the points in a table.

$\begin{array}{c} x & f (x) \\ \frac{π}{2} - 90 & 3 \\ π - 90 & 2 \\ \frac{3 π}{2} - 90 & 1 \\ 2 π - 90 & 2 \\ \frac{5 π}{2} - 90 & 3 \end{array}$

The trig function can be graphed using the amplitude, period, phase shift, vertical shift, and the points.

Amplitude: $1$

Period: $2 π$

Phase Shift: $- 90$ ( $90$ to the left)

Vertical Shift: $2$

$\begin{array}{c} x & f (x) \\ \frac{π}{2} - 90 & 3 \\ π - 90 & 2 \\ \frac{3 π}{2} - 90 & 1 \\ 2 π - 90 & 2 \\ \frac{5 π}{2} - 90 & 3 \end{array}$

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