Trigonometry Examples

(\cot (x) + 1) (\csc (x) - \frac{1}{2}) = 0

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

Step 1

Simplify

(\cot (x) + 1) (\csc (x) - \frac{1}{2})

$(\cot (x) + 1) (\csc (x) - \frac{1}{2})$ .

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Step 1.1

Expand

(\cot (x) + 1) (\csc (x) - \frac{1}{2})

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

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Step 1.1.1

Apply the distributive property.

\cot (x) (\csc (x) - \frac{1}{2}) + 1 (\csc (x) - \frac{1}{2}) = 0

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

Step 1.1.2

Apply the distributive property.

\cot (x) \csc (x) + \cot (x) (- \frac{1}{2}) + 1 (\csc (x) - \frac{1}{2}) = 0

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

Step 1.1.3

Apply the distributive property.

\cot (x) \csc (x) + \cot (x) (- \frac{1}{2}) + 1 \csc (x) + 1 (- \frac{1}{2}) = 0

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

\cot (x) \csc (x) + \cot (x) (- \frac{1}{2}) + 1 \csc (x) + 1 (- \frac{1}{2}) = 0

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

Step 1.2

Simplify each term.

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Step 1.2.1

Combine

\cot (x)

$\cot (x)$ and

\frac{1}{2}

$\frac{1}{2}$ .

\cot (x) \csc (x) - \frac{\cot (x)}{2} + 1 \csc (x) + 1 (- \frac{1}{2}) = 0

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

Step 1.2.2

Multiply

\csc (x)

$\csc (x)$ by

1

$1$ .

\cot (x) \csc (x) - \frac{\cot (x)}{2} + \csc (x) + 1 (- \frac{1}{2}) = 0

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

Step 1.2.3

Multiply

- \frac{1}{2}

$- \frac{1}{2}$ by

1

$1$ .

\cot (x) \csc (x) - \frac{\cot (x)}{2} + \csc (x) - \frac{1}{2} = 0

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

\cot (x) \csc (x) - \frac{\cot (x)}{2} + \csc (x) - \frac{1}{2} = 0

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

\cot (x) \csc (x) - \frac{\cot (x)}{2} + \csc (x) - \frac{1}{2} = 0

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

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

x = \frac{3 π}{4} + π n

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

n

$n$

Step 3