Graph cos((2n+1)*90)

$\cos ((2 n + 1) \cdot 90)$

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

$a = 1$

$b = 180$

$c = - 90$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $1$

Find the period of $\cos (180 x + 90)$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $180$ in the formula for period.

$\frac{2 π}{| 180 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $180$ is $180$ .

$\frac{2 π}{180}$

Cancel the common factor of $2$ and $180$ .

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Factor $2$ out of $2 π$ .

$\frac{2 (π)}{180}$

Cancel the common factors.

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Factor $2$ out of $180$ .

$\frac{2 π}{2 \cdot 90}$

Cancel the common factor.

$\frac{2 π}{2 \cdot 90}$

Rewrite the expression.

$\frac{π}{90}$

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: $\frac{- 90}{180}$

Cancel the common factor of $- 90$ and $180$ .

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Factor $90$ out of $- 90$ .

Phase Shift: $\frac{90 (- 1)}{180}$

Cancel the common factors.

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Factor $90$ out of $180$ .

Phase Shift: $\frac{90 \cdot - 1}{90 \cdot 2}$

Cancel the common factor.

Phase Shift: $\frac{90 \cdot - 1}{90 \cdot 2}$

Rewrite the expression.

Phase Shift: $\frac{- 1}{2}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{1}{2}$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{π}{90}$

Phase Shift: $- \frac{1}{2}$ ( $\frac{1}{2}$ to the left)

Vertical Shift: $0$

Select a few points to graph.

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Find the point at $x = - \frac{1}{2}$ .

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Replace the variable $x$ with $- \frac{1}{2}$ in the expression.

$f (- \frac{1}{2}) = \cos (180 (- \frac{1}{2}) + 90)$

Simplify the result.

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Simplify each term.

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Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (- \frac{1}{2}) = \cos (180 (\frac{- 1}{2}) + 90)$

Factor $2$ out of $180$ .

$f (- \frac{1}{2}) = \cos (2 (90) (\frac{- 1}{2}) + 90)$

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $90$ by $- 1$ .

$f (- \frac{1}{2}) = \cos (- 90 + 90)$

Add $- 90$ and $90$ .

$f (- \frac{1}{2}) = \cos (0)$

The exact value of $\cos (0)$ is $1$ .

$f (- \frac{1}{2}) = 1$

The final answer is $1$ .

$1$

Find the point at $x = \frac{π}{360} - \frac{1}{2}$ .

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Replace the variable $x$ with $\frac{π}{360} - \frac{1}{2}$ in the expression.

$f (\frac{π}{360} - \frac{1}{2}) = \cos (180 (\frac{π}{360} - \frac{1}{2}) + 90)$

Simplify the result.

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Simplify each term.

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Apply the distributive property.

$f (\frac{π}{360} - \frac{1}{2}) = \cos (180 (\frac{π}{360}) + 180 (- \frac{1}{2}) + 90)$

Cancel the common factor of $180$ .

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Factor $180$ out of $360$ .

$f (\frac{π}{360} - \frac{1}{2}) = \cos (180 (\frac{π}{180 (2)}) + 180 (- \frac{1}{2}) + 90)$

Cancel the common factor.

$f (\frac{π}{360} - \frac{1}{2}) = \cos (180 (\frac{π}{180 \cdot 2}) + 180 (- \frac{1}{2}) + 90)$

Rewrite the expression.

$f (\frac{π}{360} - \frac{1}{2}) = \cos (\frac{π}{2} + 180 (- \frac{1}{2}) + 90)$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (\frac{π}{360} - \frac{1}{2}) = \cos (\frac{π}{2} + 180 (\frac{- 1}{2}) + 90)$

Factor $2$ out of $180$ .

$f (\frac{π}{360} - \frac{1}{2}) = \cos (\frac{π}{2} + 2 (90) (\frac{- 1}{2}) + 90)$

Cancel the common factor.

$f (\frac{π}{360} - \frac{1}{2}) = \cos (\frac{π}{2} + 2 \cdot (90 (\frac{- 1}{2})) + 90)$

Rewrite the expression.

$f (\frac{π}{360} - \frac{1}{2}) = \cos (\frac{π}{2} + 90 \cdot - 1 + 90)$

Multiply $90$ by $- 1$ .

$f (\frac{π}{360} - \frac{1}{2}) = \cos (\frac{π}{2} - 90 + 90)$

Simplify by adding numbers.

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Add $- 90$ and $90$ .

$f (\frac{π}{360} - \frac{1}{2}) = \cos (\frac{π}{2} + 0)$

Add $\frac{π}{2}$ and $0$ .

$f (\frac{π}{360} - \frac{1}{2}) = \cos (\frac{π}{2})$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{π}{360} - \frac{1}{2}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{π}{180} - \frac{1}{2}$ .

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Replace the variable $x$ with $\frac{π}{180} - \frac{1}{2}$ in the expression.

$f (\frac{π}{180} - \frac{1}{2}) = \cos (180 (\frac{π}{180} - \frac{1}{2}) + 90)$

Simplify the result.

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Simplify each term.

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Apply the distributive property.

$f (\frac{π}{180} - \frac{1}{2}) = \cos (180 (\frac{π}{180}) + 180 (- \frac{1}{2}) + 90)$

Cancel the common factor of $180$ .

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Cancel the common factor.

$f (\frac{π}{180} - \frac{1}{2}) = \cos (180 (\frac{π}{180}) + 180 (- \frac{1}{2}) + 90)$

Rewrite the expression.

$f (\frac{π}{180} - \frac{1}{2}) = \cos (π + 180 (- \frac{1}{2}) + 90)$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (\frac{π}{180} - \frac{1}{2}) = \cos (π + 180 (\frac{- 1}{2}) + 90)$

Factor $2$ out of $180$ .

$f (\frac{π}{180} - \frac{1}{2}) = \cos (π + 2 (90) (\frac{- 1}{2}) + 90)$

Cancel the common factor.

$f (\frac{π}{180} - \frac{1}{2}) = \cos (π + 2 \cdot (90 (\frac{- 1}{2})) + 90)$

Rewrite the expression.

$f (\frac{π}{180} - \frac{1}{2}) = \cos (π + 90 \cdot - 1 + 90)$

Multiply $90$ by $- 1$ .

$f (\frac{π}{180} - \frac{1}{2}) = \cos (π - 90 + 90)$

Simplify by adding numbers.

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Add $- 90$ and $90$ .

$f (\frac{π}{180} - \frac{1}{2}) = \cos (π + 0)$

Add $π$ and $0$ .

$f (\frac{π}{180} - \frac{1}{2}) = \cos (π)$

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

$f (\frac{π}{180} - \frac{1}{2}) = - \cos (0)$

The exact value of $\cos (0)$ is $1$ .

$f (\frac{π}{180} - \frac{1}{2}) = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (\frac{π}{180} - \frac{1}{2}) = - 1$

The final answer is $- 1$ .

$- 1$

Find the point at $x = \frac{π}{120} - \frac{1}{2}$ .

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Replace the variable $x$ with $\frac{π}{120} - \frac{1}{2}$ in the expression.

$f (\frac{π}{120} - \frac{1}{2}) = \cos (180 (\frac{π}{120} - \frac{1}{2}) + 90)$

Simplify the result.

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Simplify each term.

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Apply the distributive property.

$f (\frac{π}{120} - \frac{1}{2}) = \cos (180 (\frac{π}{120}) + 180 (- \frac{1}{2}) + 90)$

Cancel the common factor of $60$ .

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Factor $60$ out of $180$ .

$f (\frac{π}{120} - \frac{1}{2}) = \cos (60 (3) (\frac{π}{120}) + 180 (- \frac{1}{2}) + 90)$

Factor $60$ out of $120$ .

$f (\frac{π}{120} - \frac{1}{2}) = \cos (60 \cdot (3 (\frac{π}{60 \cdot 2})) + 180 (- \frac{1}{2}) + 90)$

Cancel the common factor.

$f (\frac{π}{120} - \frac{1}{2}) = \cos (60 \cdot (3 (\frac{π}{60 \cdot 2})) + 180 (- \frac{1}{2}) + 90)$

Rewrite the expression.

$f (\frac{π}{120} - \frac{1}{2}) = \cos (3 (\frac{π}{2}) + 180 (- \frac{1}{2}) + 90)$

Combine $3$ and $\frac{π}{2}$ .

$f (\frac{π}{120} - \frac{1}{2}) = \cos (\frac{3 π}{2} + 180 (- \frac{1}{2}) + 90)$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (\frac{π}{120} - \frac{1}{2}) = \cos (\frac{3 π}{2} + 180 (\frac{- 1}{2}) + 90)$

Factor $2$ out of $180$ .

$f (\frac{π}{120} - \frac{1}{2}) = \cos (\frac{3 π}{2} + 2 (90) (\frac{- 1}{2}) + 90)$

Cancel the common factor.

$f (\frac{π}{120} - \frac{1}{2}) = \cos (\frac{3 π}{2} + 2 \cdot (90 (\frac{- 1}{2})) + 90)$

Rewrite the expression.

$f (\frac{π}{120} - \frac{1}{2}) = \cos (\frac{3 π}{2} + 90 \cdot - 1 + 90)$

Multiply $90$ by $- 1$ .

$f (\frac{π}{120} - \frac{1}{2}) = \cos (\frac{3 π}{2} - 90 + 90)$

Simplify by adding numbers.

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Add $- 90$ and $90$ .

$f (\frac{π}{120} - \frac{1}{2}) = \cos (\frac{3 π}{2} + 0)$

Add $\frac{3 π}{2}$ and $0$ .

$f (\frac{π}{120} - \frac{1}{2}) = \cos (\frac{3 π}{2})$

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

$f (\frac{π}{120} - \frac{1}{2}) = \cos (\frac{π}{2})$

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$f (\frac{π}{120} - \frac{1}{2}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{π}{90} - \frac{1}{2}$ .

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Replace the variable $x$ with $\frac{π}{90} - \frac{1}{2}$ in the expression.

$f (\frac{π}{90} - \frac{1}{2}) = \cos (180 (\frac{π}{90} - \frac{1}{2}) + 90)$

Simplify the result.

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Simplify each term.

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Apply the distributive property.

$f (\frac{π}{90} - \frac{1}{2}) = \cos (180 (\frac{π}{90}) + 180 (- \frac{1}{2}) + 90)$

Cancel the common factor of $90$ .

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Factor $90$ out of $180$ .

$f (\frac{π}{90} - \frac{1}{2}) = \cos (90 (2) (\frac{π}{90}) + 180 (- \frac{1}{2}) + 90)$

Cancel the common factor.

$f (\frac{π}{90} - \frac{1}{2}) = \cos (90 \cdot (2 (\frac{π}{90})) + 180 (- \frac{1}{2}) + 90)$

Rewrite the expression.

$f (\frac{π}{90} - \frac{1}{2}) = \cos (2 π + 180 (- \frac{1}{2}) + 90)$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (\frac{π}{90} - \frac{1}{2}) = \cos (2 π + 180 (\frac{- 1}{2}) + 90)$

Factor $2$ out of $180$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $90$ by $- 1$ .

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

Simplify by adding numbers.

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Add $- 90$ and $90$ .

$f (\frac{π}{90} - \frac{1}{2}) = \cos (2 π + 0)$

Add $2 π$ and $0$ .

$f (\frac{π}{90} - \frac{1}{2}) = \cos (2 π)$

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

$f (\frac{π}{90} - \frac{1}{2}) = \cos (0)$

The exact value of $\cos (0)$ is $1$ .

$f (\frac{π}{90} - \frac{1}{2}) = 1$

The final answer is $1$ .

$1$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{1}{2} & 1 \\ \frac{π}{360} - \frac{1}{2} & 0 \\ \frac{π}{180} - \frac{1}{2} & - 1 \\ \frac{π}{120} - \frac{1}{2} & 0 \\ \frac{π}{90} - \frac{1}{2} & 1 \end{array}$

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

Amplitude: $1$

Period: $\frac{π}{90}$

Phase Shift: $- \frac{1}{2}$ ( $\frac{1}{2}$ to the left)

Vertical Shift: $0$

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