Graph f(x)=cos(x) g(x)=cos(x+pi/2)

$f (x) = \cos (x)$ $g (x) = \cos (x + \frac{π}{2})$

Graph $f (x) = \cos (x)$ .

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

$a = 1$

$b = 1$

$c = 0$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $1$

Find the period of $\cos (x)$ .

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 $\frac{c}{b}$ .

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: $\frac{0}{1}$

Divide $0$ by $1$ .

Phase Shift: $0$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $2 π$

Phase Shift: None

Vertical Shift: None

Select a few points to graph.

Find the point at $x = 0$ .

Replace the variable $x$ with $0$ in the expression.

$f (0) = \cos (0)$

Simplify the result.

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

$f (0) = 1$

The final answer is $1$ .

$1$

Find the point at $x = \frac{π}{2}$ .

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

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

Simplify the result.

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

$f (\frac{π}{2}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = π$ .

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

$f (π) = \cos (π)$

Simplify the result.

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.

$f (π) = - \cos (0)$

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

$f (π) = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (π) = - 1$

The final answer is $- 1$ .

$- 1$

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

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

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

Simplify the result.

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

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

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

$f (\frac{3 π}{2}) = 0$

The final answer is $0$ .

$0$

Find the point at $x = 2 π$ .

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

$f (2 π) = \cos (2 π)$

Simplify the result.

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

$f (2 π) = \cos (0)$

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

$f (2 π) = 1$

The final answer is $1$ .

$1$

List the points in a table.

$\begin{array}{c} x & f (x) \\ 0 & 1 \\ \frac{π}{2} & 0 \\ π & - 1 \\ \frac{3 π}{2} & 0 \\ 2 π & 1 \end{array}$

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

Amplitude: $1$

Period: $2 π$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & 1 \\ \frac{π}{2} & 0 \\ π & - 1 \\ \frac{3 π}{2} & 0 \\ 2 π & 1 \end{array}$

Amplitude: $1$

Period: $2 π$

Phase Shift: None

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ 0 & 1 \\ \frac{π}{2} & 0 \\ π & - 1 \\ \frac{3 π}{2} & 0 \\ 2 π & 1 \end{array}$

Graph $g (x) = \cos (x + \frac{π}{2})$ .

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

$a = 1$

$b = 1$

$c = - \frac{π}{2}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $1$

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

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 $\frac{c}{b}$ .

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: $\frac{- \frac{π}{2}}{1}$

Divide $- \frac{π}{2}$ by $1$ .

Phase Shift: $- \frac{π}{2}$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $2 π$

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

Vertical Shift: None

Select a few points to graph.

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

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

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

Simplify the result.

Combine the numerators over the common denominator.

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

Add $- π$ and $π$ .

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

Divide $0$ by $2$ .

$f (- \frac{π}{2}) = \cos (0)$

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

$f (- \frac{π}{2}) = 1$

The final answer is $1$ .

$1$

Find the point at $x = 0$ .

Replace the variable $x$ with $0$ in the expression.

$f (0) = \cos ((0) + \frac{π}{2})$

Simplify the result.

Add $0$ and $\frac{π}{2}$ .

$f (0) = \cos (\frac{π}{2})$

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

$f (0) = 0$

The final answer is $0$ .

$0$

Find the point at $x = \frac{π}{2}$ .

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

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

Simplify the result.

Combine the numerators over the common denominator.

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

Add $π$ and $π$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $π$ by $1$ .

$f (\frac{π}{2}) = \cos (π)$

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.

$f (\frac{π}{2}) = - \cos (0)$

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

$f (\frac{π}{2}) = - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$f (\frac{π}{2}) = - 1$

The final answer is $- 1$ .

$- 1$

Find the point at $x = π$ .

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

$f (π) = \cos ((π) + \frac{π}{2})$

Simplify the result.

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Combine fractions.

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Move $2$ to the left of $π$ .

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

Add $2 π$ and $π$ .

$f (π) = \cos (\frac{3 π}{2})$

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

$f (π) = \cos (\frac{π}{2})$

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

$f (π) = 0$

The final answer is $0$ .

$0$

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

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

$f (\frac{3 π}{2}) = \cos ((\frac{3 π}{2}) + \frac{π}{2})$

Simplify the result.

Combine the numerators over the common denominator.

$f (\frac{3 π}{2}) = \cos (\frac{3 π + π}{2})$

Add $3 π$ and $π$ .

$f (\frac{3 π}{2}) = \cos (\frac{4 π}{2})$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 π$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2 π$ by $1$ .

$f (\frac{3 π}{2}) = \cos (2 π)$

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

$f (\frac{3 π}{2}) = \cos (0)$

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

$f (\frac{3 π}{2}) = 1$

The final answer is $1$ .

$1$

List the points in a table.

$\begin{array}{c} x & f (x) \\ - \frac{π}{2} & 1 \\ 0 & 0 \\ \frac{π}{2} & - 1 \\ π & 0 \\ \frac{3 π}{2} & 1 \end{array}$

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

Amplitude: $1$

Period: $2 π$

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

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - \frac{π}{2} & 1 \\ 0 & 0 \\ \frac{π}{2} & - 1 \\ π & 0 \\ \frac{3 π}{2} & 1 \end{array}$

Amplitude: $1$

Period: $2 π$

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

Vertical Shift: None

$\begin{array}{c} x & f (x) \\ - \frac{π}{2} & 1 \\ 0 & 0 \\ \frac{π}{2} & - 1 \\ π & 0 \\ \frac{3 π}{2} & 1 \end{array}$

Plot each graph on the same coordinate system.

$f (x) = \cos (x)$

$g (x) = \cos (x + \frac{π}{2})$

Graph f(x)=cos(x) g(x)=cos(x+pi/2)

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