Trigonometry Examples

$\tan (30) = \frac{12}{x}$

Step 1

Rewrite the equation as $\frac{12}{x} = \tan (30)$ .

$\frac{12}{x} = \tan (30)$

Step 2

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

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

Step 3

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

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

Step 4

Solve the equation for $x$ .

Tap for more steps...

Step 4.1

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

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

Step 4.2

Multiply $12$ by $3$ .

$x \sqrt{3} = 36$

Step 4.3

Divide each term in $x \sqrt{3} = 36$ by $\sqrt{3}$ and simplify.

Tap for more steps...

Step 4.3.1

Divide each term in $x \sqrt{3} = 36$ by $\sqrt{3}$ .

$\frac{x \sqrt{3}}{\sqrt{3}} = \frac{36}{\sqrt{3}}$

Step 4.3.2

Simplify the left side.

Tap for more steps...

Step 4.3.2.1

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

Tap for more steps...

Step 4.3.2.1.1

Cancel the common factor.

$\frac{x \sqrt{3}}{\sqrt{3}} = \frac{36}{\sqrt{3}}$

Step 4.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{36}{\sqrt{3}}$

Step 4.3.3

Simplify the right side.

Tap for more steps...

Step 4.3.3.1

Multiply $\frac{36}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \frac{36}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 4.3.3.2

Combine and simplify the denominator.

Tap for more steps...

Step 4.3.3.2.1

Multiply $\frac{36}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \frac{36 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.3.3.2.2

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

$x = \frac{36 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.3.3.2.3

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

$x = \frac{36 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.3.3.2.4

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

$x = \frac{36 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.3.3.2.5

Add $1$ and $1$ .

$x = \frac{36 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.3.3.2.6

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

Tap for more steps...

Step 4.3.3.2.6.1

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

$x = \frac{36 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.3.3.2.6.2

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

$x = \frac{36 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.3.3.2.6.3

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

$x = \frac{36 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.3.3.2.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.3.3.2.6.4.1

Cancel the common factor.

$x = \frac{36 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.3.3.2.6.4.2

Rewrite the expression.

$x = \frac{36 \sqrt{3}}{3^{1}}$

Step 4.3.3.2.6.5

Evaluate the exponent.

$x = \frac{36 \sqrt{3}}{3}$

Step 4.3.3.3

Cancel the common factor of $36$ and $3$ .

Tap for more steps...

Step 4.3.3.3.1

Factor $3$ out of $36 \sqrt{3}$ .

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

Step 4.3.3.3.2

Cancel the common factors.

Tap for more steps...

Step 4.3.3.3.2.1

Factor $3$ out of $3$ .

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

Step 4.3.3.3.2.2

Cancel the common factor.

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

Step 4.3.3.3.2.3

Rewrite the expression.

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

Step 4.3.3.3.2.4

Divide $12 \sqrt{3}$ by $1$ .

$x = 12 \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 12 \sqrt{3}$

Decimal Form:

$x = 20.78460969 \dots$