Trigonometry Examples

$y = 2 \cot (2 x)$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 2 \cot (2 x)$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Rewrite the equation as $2 \cot (2 x) = 0$ .

$2 \cot (2 x) = 0$

Step 1.2.2

Divide each term in $2 \cot (2 x) = 0$ by $2$ and simplify.

Tap for more steps...

Step 1.2.2.1

Divide each term in $2 \cot (2 x) = 0$ by $2$ .

$\frac{2 \cot (2 x)}{2} = \frac{0}{2}$

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{2 \cot (2 x)}{2} = \frac{0}{2}$

Step 1.2.2.2.1.2

Divide $\cot (2 x)$ by $1$ .

$\cot (2 x) = \frac{0}{2}$

Step 1.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.2.3.1

Divide $0$ by $2$ .

$\cot (2 x) = 0$

Step 1.2.3

Take the inverse cotangent of both sides of the equation to extract $x$ from inside the cotangent.

$2 x = arccot (0)$

Step 1.2.4

Simplify the right side.

Tap for more steps...

Step 1.2.4.1

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

$2 x = \frac{π}{2}$

Step 1.2.5

Divide each term in $2 x = \frac{π}{2}$ by $2$ and simplify.

Tap for more steps...

Step 1.2.5.1

Divide each term in $2 x = \frac{π}{2}$ by $2$ .

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

Step 1.2.5.2

Simplify the left side.

Tap for more steps...

Step 1.2.5.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.5.2.1.1

Cancel the common factor.

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

Step 1.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 1.2.5.3

Simplify the right side.

Tap for more steps...

Step 1.2.5.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 1.2.5.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 1.2.5.3.2.1

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

$x = \frac{π}{2 \cdot 2}$

Step 1.2.5.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 1.2.6

The cotangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$2 x = π + \frac{π}{2}$

Step 1.2.7

Solve for $x$ .

Tap for more steps...

Step 1.2.7.1

Simplify.

Tap for more steps...

Step 1.2.7.1.1

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

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

Step 1.2.7.1.2

Combine $π$ and $\frac{2}{2}$ .

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

Step 1.2.7.1.3

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 2 + π}{2}$

Step 1.2.7.1.4

Add $π \cdot 2$ and $π$ .

Tap for more steps...

Step 1.2.7.1.4.1

Reorder $π$ and $2$ .

$2 x = \frac{2 \cdot π + π}{2}$

Step 1.2.7.1.4.2

Add $2 \cdot π$ and $π$ .

$2 x = \frac{3 \cdot π}{2}$

Step 1.2.7.2

Divide each term in $2 x = \frac{3 \cdot π}{2}$ by $2$ and simplify.

Tap for more steps...

Step 1.2.7.2.1

Divide each term in $2 x = \frac{3 \cdot π}{2}$ by $2$ .

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

Step 1.2.7.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.7.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.7.2.2.1.1

Cancel the common factor.

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

Step 1.2.7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 \cdot π}{2}}{2}$

Step 1.2.7.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.7.2.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.7.2.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 1.2.7.2.3.2.1

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

$x = \frac{3 π}{2 \cdot 2}$

Step 1.2.7.2.3.2.2

Multiply $2$ by $2$ .

$x = \frac{3 π}{4}$

Step 1.2.8

Find the period of $\cot (2 x)$ .

Tap for more steps...

Step 1.2.8.1

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

$\frac{π}{| b |}$

Step 1.2.8.2

Replace $b$ with $2$ in the formula for period.

$\frac{π}{| 2 |}$

Step 1.2.8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{π}{2}$

Step 1.2.9

The period of the $\cot (2 x)$ function is $\frac{π}{2}$ so values will repeat every $\frac{π}{2}$ radians in both directions.

$x = \frac{π}{4} + \frac{π n}{2}, \frac{3 π}{4} + \frac{π n}{2}$ , for any integer $n$

Step 1.2.10

Consolidate the answers.

$x = \frac{π}{4} + \frac{π n}{2}$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{π}{4} + \frac{π n}{2}, 0)$ , for any integer $n$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 2 \cot (2 (0))$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = 2 \cot (2 (0))$

Step 2.2.2

Simplify the right side.

Tap for more steps...

Step 2.2.2.1

Simplify $2 \cot (2 (0))$ .

Tap for more steps...

Step 2.2.2.1.1

Rewrite $\cot (2 (0))$ in terms of sines and cosines.

$y = 2 \frac{\cos (2 \cdot 0)}{\sin (2 \cdot 0)}$

Step 2.2.2.1.2

Multiply $2$ by $0$ .

$y = 2 \frac{\cos (2 \cdot 0)}{\sin (0)}$

Step 2.2.2.1.3

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

$y = 2 \frac{\cos (2 \cdot 0)}{0}$

Step 2.2.2.2

The equation cannot be solved because it is undefined.

$Undefined$

Step 2.3

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $(\frac{π}{4} + \frac{π n}{2}, 0)$ , for any integer $n$

y-intercept(s): $None$

Step 4