Trigonometry Examples

$(\csc (x) - 2) (\cot (x) + 1) = 0$

Step 1

Simplify the left side of the equation.

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Step 1.1

Expand $(\csc (x) - 2) (\cot (x) + 1)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$\csc (x) (\cot (x) + 1) - 2 (\cot (x) + 1) = 0$

Step 1.1.2

Apply the distributive property.

$\csc (x) \cot (x) + \csc (x) \cdot 1 - 2 (\cot (x) + 1) = 0$

Step 1.1.3

Apply the distributive property.

$\csc (x) \cot (x) + \csc (x) \cdot 1 - 2 \cot (x) - 2 \cdot 1 = 0$

Step 1.2

Simplify each term.

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Step 1.2.1

Multiply $\csc (x)$ by $1$ .

$\csc (x) \cot (x) + \csc (x) - 2 \cot (x) - 2 \cdot 1 = 0$

Step 1.2.2

Multiply $- 2$ by $1$ .

$\csc (x) \cot (x) + \csc (x) - 2 \cot (x) - 2 = 0$

Step 2

Factor $\csc (x) \cot (x) + \csc (x) - 2 \cot (x) - 2$ .

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Step 2.1

Factor out the greatest common factor from each group.

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Step 2.1.1

Group the first two terms and the last two terms.

$\csc (x) \cot (x) + \csc (x) - 2 \cot (x) - 2 = 0$

Step 2.1.2

Factor out the greatest common factor (GCF) from each group.

$\csc (x) (\cot (x) + 1) - 2 (\cot (x) + 1) = 0$

Step 2.2

Factor the polynomial by factoring out the greatest common factor, $\cot (x) + 1$ .

$(\cot (x) + 1) (\csc (x) - 2) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\cot (x) + 1 = 0$

$\csc (x) - 2 = 0$

Step 4

Set $\cot (x) + 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\cot (x) + 1$ equal to $0$ .

$\cot (x) + 1 = 0$

Step 4.2

Solve $\cot (x) + 1 = 0$ for $x$ .

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Step 4.2.1

Subtract $1$ from both sides of the equation.

$\cot (x) = - 1$

Step 4.2.2

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

$x = arccot (- 1)$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

The exact value of $arccot (- 1)$ is $\frac{3 π}{4}$ .

$x = \frac{3 π}{4}$

Step 4.2.4

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.

$x = \frac{3 π}{4} - π$

Step 4.2.5

Simplify the expression to find the second solution.

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Step 4.2.5.1

Add $2 π$ to $\frac{3 π}{4} - π$ .

$x = \frac{3 π}{4} - π + 2 π$

Step 4.2.5.2

The resulting angle of $\frac{7 π}{4}$ is positive and coterminal with $\frac{3 π}{4} - π$ .

$x = \frac{7 π}{4}$

Step 4.2.6

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

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Step 4.2.6.1

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

$\frac{π}{| b |}$

Step 4.2.6.2

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

$\frac{π}{| 1 |}$

Step 4.2.6.3

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

$\frac{π}{1}$

Step 4.2.6.4

Divide $π$ by $1$ .

$π$

Step 4.2.7

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

$x = \frac{3 π}{4} + π n, \frac{7 π}{4} + π n$ , for any integer $n$

Step 5

Set $\csc (x) - 2$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $\csc (x) - 2$ equal to $0$ .

$\csc (x) - 2 = 0$

Step 5.2

Solve $\csc (x) - 2 = 0$ for $x$ .

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Step 5.2.1

Add $2$ to both sides of the equation.

$\csc (x) = 2$

Step 5.2.2

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

$x = arccsc (2)$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

The exact value of $arccsc (2)$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 5.2.4

The cosecant function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = π - \frac{π}{6}$

Step 5.2.5

Simplify $π - \frac{π}{6}$ .

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Step 5.2.5.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 5.2.5.2

Combine fractions.

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Step 5.2.5.2.1

Combine $π$ and $\frac{6}{6}$ .

$x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 5.2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Step 5.2.5.3

Simplify the numerator.

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Step 5.2.5.3.1

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Step 5.2.5.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

Step 5.2.6

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

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Step 5.2.6.1

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

$\frac{2 π}{| b |}$

Step 5.2.6.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.6.3

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

$\frac{2 π}{1}$

Step 5.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.7

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

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 6

The final solution is all the values that make $(\cot (x) + 1) (\csc (x) - 2) = 0$ true.

$x = \frac{3 π}{4} + π n, \frac{7 π}{4} + π n, \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 7

Consolidate $\frac{3 π}{4} + π n$ and $\frac{7 π}{4} + π n$ to $\frac{3 π}{4} + π n$ .

$x = \frac{3 π}{4} + π n, \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$