Trigonometry Examples

$y = 2 + \cot (x)$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = 2 + \cot (x)$

Step 1.2

Solve the equation.

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Step 1.2.1

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

$2 + \cot (x) = 0$

Step 1.2.2

Subtract $2$ from both sides of the equation.

$\cot (x) = - 2$

Step 1.2.3

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

$x = arccot (- 2)$

Step 1.2.4

Simplify the right side.

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Step 1.2.4.1

Evaluate $arccot (- 2)$ .

$x = 2.67794504$

Step 1.2.5

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

$x = 2.67794504 - (3.14159265)$

Step 1.2.6

Simplify the expression to find the second solution.

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Step 1.2.6.1

Add $2 π$ to $2.67794504 - (3.14159265)$ .

$x = 2.67794504 - (3.14159265) + 2 π$

Step 1.2.6.2

The resulting angle of $5.81953769$ is positive and coterminal with $2.67794504 - (3.14159265)$ .

$x = 5.81953769$

Step 1.2.7

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

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Step 1.2.7.1

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

$\frac{π}{| b |}$

Step 1.2.7.2

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

$\frac{π}{| 1 |}$

Step 1.2.7.3

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

$\frac{π}{1}$

Step 1.2.7.4

Divide $π$ by $1$ .

$π$

Step 1.2.8

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

$x = 2.67794504 + π n, 5.81953769 + π n$ , for any integer $n$

Step 1.2.9

Consolidate $2.67794504 + π n$ and $5.81953769 + π n$ to $2.67794504 + π n$ .

$x = 2.67794504 + π n$ , for any integer $n$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(2.67794504 + π n, 0)$ , for any integer $n$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = 2 + \cot (0)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 2 + \cot (0)$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $2 + \cot (0)$ .

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Step 2.2.2.1.1

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

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

Step 2.2.2.1.2

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

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

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): $(2.67794504 + π n, 0)$ , for any integer $n$

y-intercept(s): $None$

Step 4