Precalculus Examples

$\tan (\frac{5 π}{6}) = \frac{y}{x}$

Step 1

Rewrite the equation as $\frac{y}{x} = \tan (\frac{5 π}{6})$ .

$\frac{y}{x} = \tan (\frac{5 π}{6})$

Step 2

Simplify both sides.

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Step 2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$\frac{y}{x} = - \tan (\frac{π}{6})$

Step 2.2

The exact value of $\tan (\frac{π}{6})$ is $\frac{\sqrt{3}}{3}$ .

$\frac{y}{x} = - \frac{\sqrt{3}}{3}$

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.

$y \cdot 3 = x (- \sqrt{3})$

Step 4

Solve the equation for $x$ .

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Step 4.1

Rewrite the equation as $x (- \sqrt{3}) = y \cdot 3$ .

$x (- \sqrt{3}) = y \cdot 3$

Step 4.2

Divide each term in $x (- \sqrt{3}) = y \cdot 3$ by $- \sqrt{3}$ and simplify.

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Step 4.2.1

Divide each term in $x (- \sqrt{3}) = y \cdot 3$ by $- \sqrt{3}$ .

$\frac{x (- \sqrt{3})}{- \sqrt{3}} = \frac{y \cdot 3}{- \sqrt{3}}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x (\sqrt{3})}{\sqrt{3}} = \frac{y \cdot 3}{- \sqrt{3}}$

Step 4.2.2.2

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

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Step 4.2.2.2.1

Cancel the common factor.

$\frac{x \sqrt{3}}{\sqrt{3}} = \frac{y \cdot 3}{- \sqrt{3}}$

Step 4.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{y \cdot 3}{- \sqrt{3}}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Move $3$ to the left of $y$ .

$x = \frac{3 \cdot y}{- \sqrt{3}}$

Step 4.2.3.2

Move the negative in front of the fraction.

$x = - \frac{3 \cdot y}{\sqrt{3}}$

Step 4.2.3.3

Multiply $\frac{3 \cdot y}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = - (\frac{3 \cdot y}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 4.2.3.4

Combine and simplify the denominator.

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Step 4.2.3.4.1

Multiply $\frac{3 \cdot y}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = - \frac{3 \cdot y \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 4.2.3.4.2

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

$x = - \frac{3 \cdot y \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.2.3.4.3

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

$x = - \frac{3 \cdot y \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.2.3.4.4

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

$x = - \frac{3 \cdot y \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.2.3.4.5

Add $1$ and $1$ .

$x = - \frac{3 \cdot y \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.2.3.4.6

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

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Step 4.2.3.4.6.1

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

$x = - \frac{3 \cdot y \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 4.2.3.4.6.2

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

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

Step 4.2.3.4.6.3

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

$x = - \frac{3 \cdot y \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.2.3.4.6.4

Cancel the common factor of $2$ .

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Step 4.2.3.4.6.4.1

Cancel the common factor.

$x = - \frac{3 \cdot y \sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.2.3.4.6.4.2

Rewrite the expression.

$x = - \frac{3 \cdot y \sqrt{3}}{3^{1}}$

Step 4.2.3.4.6.5

Evaluate the exponent.

$x = - \frac{3 \cdot y \sqrt{3}}{3}$

Step 4.2.3.5

Cancel the common factor of $3$ .

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Step 4.2.3.5.1

Cancel the common factor.

$x = - \frac{3 \cdot y \sqrt{3}}{3}$

Step 4.2.3.5.2

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

$x = - (y \sqrt{3})$

$x = - y \sqrt{3}$