Trigonometry Examples

$\tan (2 x) = \frac{2 (\frac{12}{5})}{1 - {(\frac{12}{5})}^{2}}$

Step 1

Subtract $\frac{2 (\frac{12}{5})}{1 - {(\frac{12}{5})}^{2}}$ from both sides of the equation.

$\tan (2 x) - \frac{2 (\frac{12}{5})}{1 - {(\frac{12}{5})}^{2}} = 0$

Step 2

Simplify each term.

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Step 2.1

Combine $2$ and $\frac{12}{5}$ .

$\tan (2 x) - \frac{\frac{2 \cdot 12}{5}}{1 - {(\frac{12}{5})}^{2}} = 0$

Step 2.2

Simplify the denominator.

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Step 2.2.1

Apply the product rule to $\frac{12}{5}$ .

$\tan (2 x) - \frac{\frac{2 \cdot 12}{5}}{1 - \frac{12^{2}}{5^{2}}} = 0$

Step 2.2.2

Raise $12$ to the power of $2$ .

$\tan (2 x) - \frac{\frac{2 \cdot 12}{5}}{1 - \frac{144}{5^{2}}} = 0$

Step 2.2.3

Raise $5$ to the power of $2$ .

$\tan (2 x) - \frac{\frac{2 \cdot 12}{5}}{1 - \frac{144}{25}} = 0$

Step 2.2.4

Write $1$ as a fraction with a common denominator.

$\tan (2 x) - \frac{\frac{2 \cdot 12}{5}}{\frac{25}{25} - \frac{144}{25}} = 0$

Step 2.2.5

Combine the numerators over the common denominator.

$\tan (2 x) - \frac{\frac{2 \cdot 12}{5}}{\frac{25 - 144}{25}} = 0$

Step 2.2.6

Subtract $144$ from $25$ .

$\tan (2 x) - \frac{\frac{2 \cdot 12}{5}}{\frac{- 119}{25}} = 0$

Step 2.2.7

Move the negative in front of the fraction.

$\tan (2 x) - \frac{\frac{2 \cdot 12}{5}}{- \frac{119}{25}} = 0$

Step 2.3

Multiply $2$ by $12$ .

$\tan (2 x) - \frac{\frac{24}{5}}{- \frac{119}{25}} = 0$

Step 2.4

Multiply the numerator by the reciprocal of the denominator.

$\tan (2 x) - (\frac{24}{5} (- \frac{25}{119})) = 0$

Step 2.5

Cancel the common factor of $5$ .

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Step 2.5.1

Move the leading negative in $- \frac{25}{119}$ into the numerator.

$\tan (2 x) - (\frac{24}{5} \cdot \frac{- 25}{119}) = 0$

Step 2.5.2

Factor $5$ out of $- 25$ .

$\tan (2 x) - (\frac{24}{5} \cdot \frac{5 (- 5)}{119}) = 0$

Step 2.5.3

Cancel the common factor.

$\tan (2 x) - (\frac{24}{5} \cdot \frac{5 \cdot - 5}{119}) = 0$

Step 2.5.4

Rewrite the expression.

$\tan (2 x) - (24 (\frac{- 5}{119})) = 0$

Step 2.6

Combine $24$ and $\frac{- 5}{119}$ .

$\tan (2 x) - \frac{24 \cdot - 5}{119} = 0$

Step 2.7

Multiply $24$ by $- 5$ .

$\tan (2 x) - \frac{- 120}{119} = 0$

Step 2.8

Move the negative in front of the fraction.

$\tan (2 x) - - \frac{120}{119} = 0$

Step 2.9

Multiply $- - \frac{120}{119}$ .

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Step 2.9.1

Multiply $- 1$ by $- 1$ .

$\tan (2 x) + 1 (\frac{120}{119}) = 0$

Step 2.9.2

Multiply $\frac{120}{119}$ by $1$ .

$\tan (2 x) + \frac{120}{119} = 0$

Step 3

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

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

Step 4

Divide each term in $2 x = \frac{π}{2} + π n$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = \frac{π}{2} + π n$ by $2$ .

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $x$ by $1$ .

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

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.3.1.2

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Step 4.3.1.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{2}$ .

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

Step 4.3.1.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4} + \frac{π n}{2}$

Step 5

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 $0$ .

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

Step 6