Trigonometry Examples

$\tan (60) = \frac{12}{x}$

Step 1

Rewrite the equation as $\frac{12}{x} = \tan (60)$ .

$\frac{12}{x} = \tan (60)$

Step 2

The exact value of $\tan (60)$ is $\sqrt{3}$ .

$\frac{12}{x} = \sqrt{3}$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 3.2

The LCM of one and any expression is the expression.

$x$

Step 4

Multiply each term in $\frac{12}{x} = \sqrt{3}$ by $x$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{12}{x} = \sqrt{3}$ by $x$ .

$\frac{12}{x} x = \sqrt{3} x$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $x$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{12}{x} x = \sqrt{3} x$

Step 4.2.1.2

Rewrite the expression.

$12 = \sqrt{3} x$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $\sqrt{3} x = 12$ .

$\sqrt{3} x = 12$

Step 5.2

Divide each term in $\sqrt{3} x = 12$ by $\sqrt{3}$ and simplify.

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Step 5.2.1

Divide each term in $\sqrt{3} x = 12$ by $\sqrt{3}$ .

$\frac{\sqrt{3} x}{\sqrt{3}} = \frac{12}{\sqrt{3}}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $\sqrt{3}$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{\sqrt{3} x}{\sqrt{3}} = \frac{12}{\sqrt{3}}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{\sqrt{3}}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Multiply $\frac{12}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \frac{12}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5.2.3.2

Combine and simplify the denominator.

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Step 5.2.3.2.1

Multiply $\frac{12}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \frac{12 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.2.3.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{12 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 5.2.3.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \frac{12 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 5.2.3.2.4

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

$x = \frac{12 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 5.2.3.2.5

Add $1$ and $1$ .

$x = \frac{12 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 5.2.3.2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.2.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \frac{12 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 5.2.3.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{12 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 5.2.3.2.6.3

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

$x = \frac{12 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.2.3.2.6.4

Cancel the common factor of $2$ .

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Step 5.2.3.2.6.4.1

Cancel the common factor.

$x = \frac{12 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.2.3.2.6.4.2

Rewrite the expression.

$x = \frac{12 \sqrt{3}}{3^{1}}$

Step 5.2.3.2.6.5

Evaluate the exponent.

$x = \frac{12 \sqrt{3}}{3}$

Step 5.2.3.3

Cancel the common factor of $12$ and $3$ .

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Step 5.2.3.3.1

Factor $3$ out of $12 \sqrt{3}$ .

$x = \frac{3 (4 \sqrt{3})}{3}$

Step 5.2.3.3.2

Cancel the common factors.

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Step 5.2.3.3.2.1

Factor $3$ out of $3$ .

$x = \frac{3 (4 \sqrt{3})}{3 (1)}$

Step 5.2.3.3.2.2

Cancel the common factor.

$x = \frac{3 (4 \sqrt{3})}{3 \cdot 1}$

Step 5.2.3.3.2.3

Rewrite the expression.

$x = \frac{4 \sqrt{3}}{1}$

Step 5.2.3.3.2.4

Divide $4 \sqrt{3}$ by $1$ .

$x = 4 \sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 4 \sqrt{3}$

Decimal Form:

$x = 6.92820323 \dots$