Trigonometry Examples

$y = 3 \tan (\frac{x}{4} \cdot x + \frac{π}{2})$

Step 1

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

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

Step 2

Solve for $x$ .

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Step 2.1

Multiply $\frac{x}{4} x$ .

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Step 2.1.1

Combine $\frac{x}{4}$ and $x$ .

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

Step 2.1.2

Raise $x$ to the power of $1$ .

$\frac{x^{1} x}{4} + \frac{π}{2} = \frac{π}{2} + π n$

Step 2.1.3

Raise $x$ to the power of $1$ .

$\frac{x^{1} x^{1}}{4} + \frac{π}{2} = \frac{π}{2} + π n$

Step 2.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{1 + 1}}{4} + \frac{π}{2} = \frac{π}{2} + π n$

Step 2.1.5

Add $1$ and $1$ .

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

Step 2.2

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

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Step 2.2.1

Subtract $\frac{π}{2}$ from both sides of the equation.

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

Step 2.2.2

Combine the opposite terms in $\frac{π}{2} + π n - \frac{π}{2}$ .

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Step 2.2.2.1

Combine the numerators over the common denominator.

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

Step 2.2.2.2

Subtract $π$ from $π$ .

$\frac{x^{2}}{4} = π n + \frac{0}{2}$

Step 2.2.3

Divide $0$ by $2$ .

$\frac{x^{2}}{4} = π n + 0$

Step 2.2.4

Add $π n$ and $0$ .

$\frac{x^{2}}{4} = π n$

Step 2.3

Multiply both sides of the equation by $4$ .

$4 \frac{x^{2}}{4} = 4 (π n)$

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 $4$ .

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Step 2.4.1.1.1

Cancel the common factor.

$4 \frac{x^{2}}{4} = 4 (π n)$

Step 2.4.1.1.2

Rewrite the expression.

$x^{2} = 4 (π n)$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Remove parentheses.

$x^{2} = 4 π n$

Step 2.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4 π n}$

Step 2.6

Simplify $\pm \sqrt{4 π n}$ .

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Step 2.6.1

Rewrite $4 π n$ as $2^{2} (π n)$ .

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Step 2.6.1.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2} π n}$

Step 2.6.1.2

Add parentheses.

$x = \pm \sqrt{2^{2} (π n)}$

Step 2.6.2

Pull terms out from under the radical.

$x = \pm 2 \sqrt{π n}$

Step 2.7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.7.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2 \sqrt{π n}$

Step 2.7.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2 \sqrt{π n}$

Step 2.7.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2 \sqrt{π n}, - 2 \sqrt{π n}$

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 $0$ .

${x | x = 2 \sqrt{π n}, x = - 2 \sqrt{π n}}$ $n$ , for any integer $n$

Step 4