Trigonometry Examples

$y = 2 \tan (\frac{1}{2} x - 3 π)$

Step 1

Set the argument in $\tan (\frac{1}{2} x - 3 π)$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$\frac{1}{2} x - 3 π = \frac{π}{2} + π n$ , for any integer $n$

Step 2

Solve for $x$ .

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Step 2.1

Combine $\frac{1}{2}$ and $x$ .

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

Step 2.2

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

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Step 2.2.1

Add $3 π$ to both sides of the equation.

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

Step 2.2.2

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

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

Step 2.2.3

Combine $3 π$ and $\frac{2}{2}$ .

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

Step 2.2.4

Combine the numerators over the common denominator.

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

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $2$ by $3$ .

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

Step 2.2.5.2

Add $π$ and $6 π$ .

$\frac{x}{2} = π n + \frac{7 π}{2}$

Step 2.3

Multiply both sides of the equation by $2$ .

$2 \frac{x}{2} = 2 (π n + \frac{7 π}{2})$

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

Cancel the common factor of $2$ .

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Step 2.4.1.1.1

Cancel the common factor.

$2 \frac{x}{2} = 2 (π n + \frac{7 π}{2})$

Step 2.4.1.1.2

Rewrite the expression.

$x = 2 (π n + \frac{7 π}{2})$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify $2 (π n + \frac{7 π}{2})$ .

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Step 2.4.2.1.1

Apply the distributive property.

$x = 2 (π n) + 2 \frac{7 π}{2}$

Step 2.4.2.1.2

Cancel the common factor of $2$ .

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Step 2.4.2.1.2.1

Cancel the common factor.

$x = 2 π n + 2 \frac{7 π}{2}$

Step 2.4.2.1.2.2

Rewrite the expression.

$x = 2 π n + 7 π$

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 $0$ .

${x | x = 2 π n + 7 π}$ $n$ , for any integer $n$

Step 4