Trigonometry Examples

$f (x) = \cot (\frac{3}{2} x + \frac{π}{4}) - 2$

Step 1

Find the asymptotes.

Tap for more steps...

Step 1.1

For any $y = \cot (x)$ , vertical asymptotes occur at $x = n π$ , where $n$ is an integer. Use the basic period for $y = \cot (x)$ , $(0, π)$ , to find the vertical asymptotes for $y = \cot (\frac{3 x}{2} + \frac{π}{4}) - 2$ . Set the inside of the cotangent function, $b x + c$ , for $y = a \cot (b x + c) + d$ equal to $0$ to find where the vertical asymptote occurs for $y = \cot (\frac{3 x}{2} + \frac{π}{4}) - 2$ .

$\frac{3 x}{2} + \frac{π}{4} = 0$

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Subtract $\frac{π}{4}$ from both sides of the equation.

$\frac{3 x}{2} = - \frac{π}{4}$

Step 1.2.2

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} (- \frac{π}{4})$

Step 1.2.3

Simplify both sides of the equation.

Tap for more steps...

Step 1.2.3.1

Simplify the left side.

Tap for more steps...

Step 1.2.3.1.1

Simplify $\frac{2}{3} \cdot \frac{3 x}{2}$ .

Tap for more steps...

Step 1.2.3.1.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.3.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} (- \frac{π}{4})$

Step 1.2.3.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{2}{3} (- \frac{π}{4})$

Step 1.2.3.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.2.3.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{2}{3} (- \frac{π}{4})$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{2}{3} (- \frac{π}{4})$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{2}{3} (- \frac{π}{4})$

Step 1.2.3.2

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.1

Simplify $\frac{2}{3} (- \frac{π}{4})$ .

Tap for more steps...

Step 1.2.3.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.3.2.1.1.1

Move the leading negative in $- \frac{π}{4}$ into the numerator.

$x = \frac{2}{3} \cdot \frac{- π}{4}$

Step 1.2.3.2.1.1.2

Factor $2$ out of $4$ .

$x = \frac{2}{3} \cdot \frac{- π}{2 (2)}$

Step 1.2.3.2.1.1.3

Cancel the common factor.

$x = \frac{2}{3} \cdot \frac{- π}{2 \cdot 2}$

Step 1.2.3.2.1.1.4

Rewrite the expression.

$x = \frac{1}{3} \cdot \frac{- π}{2}$

Step 1.2.3.2.1.2

Multiply $\frac{1}{3}$ by $\frac{- π}{2}$ .

$x = \frac{- π}{3 \cdot 2}$

Step 1.2.3.2.1.3

Simplify the expression.

Tap for more steps...

Step 1.2.3.2.1.3.1

Multiply $3$ by $2$ .

$x = \frac{- π}{6}$

Step 1.2.3.2.1.3.2

Move the negative in front of the fraction.

$x = - \frac{π}{6}$

Step 1.3

Set the inside of the cotangent function $\frac{3 x}{2} + \frac{π}{4}$ equal to $π$ .

$\frac{3 x}{2} + \frac{π}{4} = π$

Step 1.4

Solve for $x$ .

Tap for more steps...

Step 1.4.1

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 1.4.1.1

Subtract $\frac{π}{4}$ from both sides of the equation.

$\frac{3 x}{2} = π - \frac{π}{4}$

Step 1.4.1.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{3 x}{2} = π \cdot \frac{4}{4} - \frac{π}{4}$

Step 1.4.1.3

Combine $π$ and $\frac{4}{4}$ .

$\frac{3 x}{2} = \frac{π \cdot 4}{4} - \frac{π}{4}$

Step 1.4.1.4

Combine the numerators over the common denominator.

$\frac{3 x}{2} = \frac{π \cdot 4 - π}{4}$

Step 1.4.1.5

Simplify the numerator.

Tap for more steps...

Step 1.4.1.5.1

Move $4$ to the left of $π$ .

$\frac{3 x}{2} = \frac{4 \cdot π - π}{4}$

Step 1.4.1.5.2

Subtract $π$ from $4 π$ .

$\frac{3 x}{2} = \frac{3 π}{4}$

Step 1.4.2

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \cdot \frac{3 π}{4}$

Step 1.4.3

Simplify both sides of the equation.

Tap for more steps...

Step 1.4.3.1

Simplify the left side.

Tap for more steps...

Step 1.4.3.1.1

Simplify $\frac{2}{3} \cdot \frac{3 x}{2}$ .

Tap for more steps...

Step 1.4.3.1.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.3.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 x}{2} = \frac{2}{3} \cdot \frac{3 π}{4}$

Step 1.4.3.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 x) = \frac{2}{3} \cdot \frac{3 π}{4}$

Step 1.4.3.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.4.3.1.1.2.1

Factor $3$ out of $3 x$ .

$\frac{1}{3} (3 (x)) = \frac{2}{3} \cdot \frac{3 π}{4}$

Step 1.4.3.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 x) = \frac{2}{3} \cdot \frac{3 π}{4}$

Step 1.4.3.1.1.2.3

Rewrite the expression.

$x = \frac{2}{3} \cdot \frac{3 π}{4}$

Step 1.4.3.2

Simplify the right side.

Tap for more steps...

Step 1.4.3.2.1

Simplify $\frac{2}{3} \cdot \frac{3 π}{4}$ .

Tap for more steps...

Step 1.4.3.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.4.3.2.1.1.1

Factor $2$ out of $4$ .

$x = \frac{2}{3} \cdot \frac{3 π}{2 (2)}$

Step 1.4.3.2.1.1.2

Cancel the common factor.

$x = \frac{2}{3} \cdot \frac{3 π}{2 \cdot 2}$

Step 1.4.3.2.1.1.3

Rewrite the expression.

$x = \frac{1}{3} \cdot \frac{3 π}{2}$

Step 1.4.3.2.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 1.4.3.2.1.2.1

Factor $3$ out of $3 π$ .

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

Step 1.4.3.2.1.2.2

Cancel the common factor.

$x = \frac{1}{3} \cdot \frac{3 π}{2}$

Step 1.4.3.2.1.2.3

Rewrite the expression.

$x = \frac{π}{2}$

Step 1.5

The basic period for $y = \cot (\frac{3 x}{2} + \frac{π}{4}) - 2$ will occur at $(- \frac{π}{6}, \frac{π}{2})$ , where $- \frac{π}{6}$ and $\frac{π}{2}$ are vertical asymptotes.

$(- \frac{π}{6}, \frac{π}{2})$

Step 1.6

Find the period $\frac{π}{| b |}$ to find where the vertical asymptotes exist.

Tap for more steps...

Step 1.6.1

$\frac{3}{2}$ is approximately $1.5$ which is positive so remove the absolute value

$\frac{π}{\frac{3}{2}}$

Step 1.6.2

Multiply the numerator by the reciprocal of the denominator.

$π \frac{2}{3}$

Step 1.6.3

Combine $π$ and $\frac{2}{3}$ .

$\frac{π \cdot 2}{3}$

Step 1.6.4

Move $2$ to the left of $π$ .

$\frac{2 π}{3}$

Step 1.7

The vertical asymptotes for $y = \cot (\frac{3 x}{2} + \frac{π}{4}) - 2$ occur at $- \frac{π}{6}$ , $\frac{π}{2}$ , and every $x = - \frac{π}{6} + \frac{2 π n}{3}$ , where $n$ is an integer.

$x = - \frac{π}{6} + \frac{2 π n}{3}$

Step 1.8

Cotangent only has vertical asymptotes.

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - \frac{π}{6} + \frac{2 π n}{3}$ where $n$ is an integer

No Horizontal Asymptotes

No Oblique Asymptotes

Vertical Asymptotes: $x = - \frac{π}{6} + \frac{2 π n}{3}$ where $n$ is an integer

Step 2

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

$a = 1$

$b = \frac{3}{2}$

$c = - \frac{π}{4}$

$d = - 2$

Step 3

Since the graph of the function $c o t$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Step 4

Find the period using the formula $\frac{π}{| b |}$ .

Tap for more steps...

Step 4.1

Find the period of $\cot (\frac{3 x}{2} + \frac{π}{4})$ .

Tap for more steps...

Step 4.1.1

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

$\frac{π}{| b |}$

Step 4.1.2

Replace $b$ with $\frac{3}{2}$ in the formula for period.

$\frac{π}{| \frac{3}{2} |}$

Step 4.1.3

$\frac{3}{2}$ is approximately $1.5$ which is positive so remove the absolute value

$\frac{π}{\frac{3}{2}}$

Step 4.1.4

Multiply the numerator by the reciprocal of the denominator.

$π \frac{2}{3}$

Step 4.1.5

Combine $π$ and $\frac{2}{3}$ .

$\frac{π \cdot 2}{3}$

Step 4.1.6

Move $2$ to the left of $π$ .

$\frac{2 π}{3}$

Step 4.2

Find the period of $- 2$ .

Tap for more steps...

Step 4.2.1

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

$\frac{π}{| b |}$

Step 4.2.2

Replace $b$ with $\frac{3}{2}$ in the formula for period.

$\frac{π}{| \frac{3}{2} |}$

Step 4.2.3

$\frac{3}{2}$ is approximately $1.5$ which is positive so remove the absolute value

$\frac{π}{\frac{3}{2}}$

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

$π \frac{2}{3}$

Step 4.2.5

Combine $π$ and $\frac{2}{3}$ .

$\frac{π \cdot 2}{3}$

Step 4.2.6

Move $2$ to the left of $π$ .

$\frac{2 π}{3}$

Step 4.3

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$\frac{2 π}{3}$

Step 5

Find the phase shift using the formula $\frac{c}{b}$ .

Tap for more steps...

Step 5.1

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Step 5.2

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{- \frac{π}{4}}{\frac{3}{2}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{4} \cdot \frac{2}{3}$

Step 5.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.4.1

Move the leading negative in $- \frac{π}{4}$ into the numerator.

Phase Shift: $\frac{- π}{4} \cdot \frac{2}{3}$

Step 5.4.2

Factor $2$ out of $4$ .

Phase Shift: $\frac{- π}{2 (2)} \cdot \frac{2}{3}$

Step 5.4.3

Cancel the common factor.

Phase Shift: $\frac{- π}{2 \cdot 2} \cdot \frac{2}{3}$

Step 5.4.4

Rewrite the expression.

Phase Shift: $\frac{- π}{2} \cdot \frac{1}{3}$

Step 5.5

Multiply $\frac{- π}{2}$ by $\frac{1}{3}$ .

Phase Shift: $\frac{- π}{2 \cdot 3}$

Step 5.6

Simplify the expression.

Tap for more steps...

Step 5.6.1

Multiply $2$ by $3$ .

Phase Shift: $\frac{- π}{6}$

Step 5.6.2

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{6}$

Step 6

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{2 π}{3}$

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

Vertical Shift: $- 2$

Step 7

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

Vertical Asymptotes: $x = - \frac{π}{6} + \frac{2 π n}{3}$ where $n$ is an integer

Amplitude: None

Period: $\frac{2 π}{3}$

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

Vertical Shift: $- 2$

Step 8