Trigonometry Examples

$\tan (\frac{π}{4} - x)$

Step 1

Set the argument in $\tan (\frac{π}{4} - x)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

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

Step 2

Solve for $x$ .

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Step 2.1

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

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Step 2.1.1

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

$- x = \frac{π}{2} + π n - \frac{π}{4}$

Step 2.1.2

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

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

Step 2.1.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.1.3.1

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

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

Step 2.1.3.2

Multiply $2$ by $2$ .

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

Step 2.1.4

Combine the numerators over the common denominator.

$- x = π n + \frac{π \cdot 2 - π}{4}$

Step 2.1.5

Subtract $π$ from $π \cdot 2$ .

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Step 2.1.5.1

Reorder $π$ and $2$ .

$- x = π n + \frac{2 \cdot π - π}{4}$

Step 2.1.5.2

Subtract $π$ from $2 \cdot π$ .

$- x = π n + \frac{π}{4}$

Step 2.2

Divide each term in $- x = π n + \frac{π}{4}$ by $- 1$ and simplify.

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Step 2.2.1

Divide each term in $- x = π n + \frac{π}{4}$ by $- 1$ .

$\frac{- x}{- 1} = \frac{π n}{- 1} + \frac{\frac{π}{4}}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{π n}{- 1} + \frac{\frac{π}{4}}{- 1}$

Step 2.2.2.2

Divide $x$ by $1$ .

$x = \frac{π n}{- 1} + \frac{\frac{π}{4}}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Move the negative one from the denominator of $\frac{π n}{- 1}$ .

$x = - 1 \cdot (π n) + \frac{\frac{π}{4}}{- 1}$

Step 2.2.3.1.2

Rewrite $- 1 \cdot (π n)$ as $- (π n)$ .

$x = - (π n) + \frac{\frac{π}{4}}{- 1}$

Step 2.2.3.1.3

Move the negative one from the denominator of $\frac{\frac{π}{4}}{- 1}$ .

$x = - π n - 1 \cdot \frac{π}{4}$

Step 2.2.3.1.4

Rewrite $- 1 \cdot \frac{π}{4}$ as $- \frac{π}{4}$ .

$x = - π n - \frac{π}{4}$

Step 3

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

${x | x = - π n - \frac{π}{4}}$ $n$ , for any integer $n$

Step 4