Trigonometry Examples

$y = 4 \tan (x - \frac{π}{3})$

Step 1

Set the argument in $\tan (x - \frac{π}{3})$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

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

Step 2

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

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Step 2.1

Add $\frac{π}{3}$ to both sides of the equation.

$x = \frac{π}{2} + π n + \frac{π}{3}$

Step 2.2

To write $\frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 2.3

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

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

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

Step 2.4.2

Multiply $2$ by $3$ .

$x = π n + \frac{π \cdot 3}{6} + \frac{π}{3} \cdot \frac{2}{2}$

Step 2.4.3

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

$x = π n + \frac{π \cdot 3}{6} + \frac{π \cdot 2}{3 \cdot 2}$

Step 2.4.4

Multiply $3$ by $2$ .

$x = π n + \frac{π \cdot 3}{6} + \frac{π \cdot 2}{6}$

Step 2.5

Combine the numerators over the common denominator.

$x = π n + \frac{π \cdot 3 + π \cdot 2}{6}$

Step 2.6

Reorder $π$ and $2$ .

$x = π n + \frac{3 \cdot π + 2 \cdot π}{6}$

Step 2.7

Add $3 \cdot π$ and $2 \cdot π$ .

$x = π n + \frac{5 π}{6}$

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 $0$ .

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

Step 4