Trigonometry Examples

$\tan (75) = \frac{20}{x}$

Step 1

Rewrite the equation as $\frac{20}{x} = \tan (75)$ .

$\frac{20}{x} = \tan (75)$

Step 2

The exact value of $\tan (75)$ is $2 + \sqrt{3}$ .

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Step 2.1

Split $75$ into two angles where the values of the six trigonometric functions are known.

$\frac{20}{x} = \tan (30 + 45)$

Step 2.2

Apply the sum of angles identity.

$\frac{20}{x} = \frac{\tan (30) + \tan (45)}{1 - \tan (30) \tan (45)}$

Step 2.3

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

$\frac{20}{x} = \frac{\frac{\sqrt{3}}{3} + \tan (45)}{1 - \tan (30) \tan (45)}$

Step 2.4

The exact value of $\tan (45)$ is $1$ .

$\frac{20}{x} = \frac{\frac{\sqrt{3}}{3} + 1}{1 - \tan (30) \tan (45)}$

Step 2.5

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

$\frac{20}{x} = \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \tan (45)}$

Step 2.6

The exact value of $\tan (45)$ is $1$ .

$\frac{20}{x} = \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$

Step 2.7

Simplify $\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$ .

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Step 2.7.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 2.7.1.1

Multiply $\frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$ by $\frac{3}{3}$ .

$\frac{20}{x} = \frac{3}{3} \cdot \frac{\frac{\sqrt{3}}{3} + 1}{1 - \frac{\sqrt{3}}{3} \cdot 1}$

Step 2.7.1.2

Combine.

$\frac{20}{x} = \frac{3 (\frac{\sqrt{3}}{3} + 1)}{3 (1 - \frac{\sqrt{3}}{3} \cdot 1)}$

Step 2.7.2

Apply the distributive property.

$\frac{20}{x} = \frac{3 \frac{\sqrt{3}}{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}$

Step 2.7.3

Cancel the common factor of $3$ .

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Step 2.7.3.1

Cancel the common factor.

$\frac{20}{x} = \frac{3 \frac{\sqrt{3}}{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}$

Step 2.7.3.2

Rewrite the expression.

$\frac{20}{x} = \frac{\sqrt{3} + 3 \cdot 1}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}$

Step 2.7.4

Multiply $3$ by $1$ .

$\frac{20}{x} = \frac{\sqrt{3} + 3}{3 \cdot 1 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}$

Step 2.7.5

Simplify the denominator.

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Step 2.7.5.1

Multiply $3$ by $1$ .

$\frac{20}{x} = \frac{\sqrt{3} + 3}{3 + 3 (- \frac{\sqrt{3}}{3} \cdot 1)}$

Step 2.7.5.2

Multiply $- 1$ by $1$ .

$\frac{20}{x} = \frac{\sqrt{3} + 3}{3 + 3 (- \frac{\sqrt{3}}{3})}$

Step 2.7.5.3

Cancel the common factor of $3$ .

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Step 2.7.5.3.1

Move the leading negative in $- \frac{\sqrt{3}}{3}$ into the numerator.

$\frac{20}{x} = \frac{\sqrt{3} + 3}{3 + 3 \frac{- \sqrt{3}}{3}}$

Step 2.7.5.3.2

Cancel the common factor.

$\frac{20}{x} = \frac{\sqrt{3} + 3}{3 + 3 \frac{- \sqrt{3}}{3}}$

Step 2.7.5.3.3

Rewrite the expression.

$\frac{20}{x} = \frac{\sqrt{3} + 3}{3 - \sqrt{3}}$

Step 2.7.6

Multiply $\frac{\sqrt{3} + 3}{3 - \sqrt{3}}$ by $\frac{3 + \sqrt{3}}{3 + \sqrt{3}}$ .

$\frac{20}{x} = \frac{\sqrt{3} + 3}{3 - \sqrt{3}} \cdot \frac{3 + \sqrt{3}}{3 + \sqrt{3}}$

Step 2.7.7

Multiply $\frac{\sqrt{3} + 3}{3 - \sqrt{3}}$ by $\frac{3 + \sqrt{3}}{3 + \sqrt{3}}$ .

$\frac{20}{x} = \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{(3 - \sqrt{3}) (3 + \sqrt{3})}$

Step 2.7.8

Expand the denominator using the FOIL method.

$\frac{20}{x} = \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{9 + 3 \sqrt{3} - 3 \sqrt{3} - {\sqrt{3}}^{2}}$

Step 2.7.9

Simplify.

$\frac{20}{x} = \frac{(\sqrt{3} + 3) (3 + \sqrt{3})}{6}$

Step 2.7.10

Simplify the numerator.

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Step 2.7.10.1

Reorder terms.

$\frac{20}{x} = \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6}$

Step 2.7.10.2

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

$\frac{20}{x} = \frac{{(3 + \sqrt{3})}^{1} (3 + \sqrt{3})}{6}$

Step 2.7.10.3

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

$\frac{20}{x} = \frac{{(3 + \sqrt{3})}^{1} {(3 + \sqrt{3})}^{1}}{6}$

Step 2.7.10.4

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

$\frac{20}{x} = \frac{{(3 + \sqrt{3})}^{1 + 1}}{6}$

Step 2.7.10.5

Add $1$ and $1$ .

$\frac{20}{x} = \frac{{(3 + \sqrt{3})}^{2}}{6}$

Step 2.7.11

Rewrite ${(3 + \sqrt{3})}^{2}$ as $(3 + \sqrt{3}) (3 + \sqrt{3})$ .

$\frac{20}{x} = \frac{(3 + \sqrt{3}) (3 + \sqrt{3})}{6}$

Step 2.7.12

Expand $(3 + \sqrt{3}) (3 + \sqrt{3})$ using the FOIL Method.

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Step 2.7.12.1

Apply the distributive property.

$\frac{20}{x} = \frac{3 (3 + \sqrt{3}) + \sqrt{3} (3 + \sqrt{3})}{6}$

Step 2.7.12.2

Apply the distributive property.

$\frac{20}{x} = \frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} (3 + \sqrt{3})}{6}$

Step 2.7.12.3

Apply the distributive property.

$\frac{20}{x} = \frac{3 \cdot 3 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6}$

Step 2.7.13

Simplify and combine like terms.

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Step 2.7.13.1

Simplify each term.

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Step 2.7.13.1.1

Multiply $3$ by $3$ .

$\frac{20}{x} = \frac{9 + 3 \sqrt{3} + \sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6}$

Step 2.7.13.1.2

Move $3$ to the left of $\sqrt{3}$ .

$\frac{20}{x} = \frac{9 + 3 \sqrt{3} + 3 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{6}$

Step 2.7.13.1.3

Combine using the product rule for radicals.

$\frac{20}{x} = \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3 \cdot 3}}{6}$

Step 2.7.13.1.4

Multiply $3$ by $3$ .

$\frac{20}{x} = \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{9}}{6}$

Step 2.7.13.1.5

Rewrite $9$ as $3^{2}$ .

$\frac{20}{x} = \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + \sqrt{3^{2}}}{6}$

Step 2.7.13.1.6

Pull terms out from under the radical, assuming positive real numbers.

$\frac{20}{x} = \frac{9 + 3 \sqrt{3} + 3 \sqrt{3} + 3}{6}$

Step 2.7.13.2

Add $9$ and $3$ .

$\frac{20}{x} = \frac{12 + 3 \sqrt{3} + 3 \sqrt{3}}{6}$

Step 2.7.13.3

Add $3 \sqrt{3}$ and $3 \sqrt{3}$ .

$\frac{20}{x} = \frac{12 + 6 \sqrt{3}}{6}$

Step 2.7.14

Cancel the common factor of $12 + 6 \sqrt{3}$ and $6$ .

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Step 2.7.14.1

Factor $6$ out of $12$ .

$\frac{20}{x} = \frac{6 \cdot 2 + 6 \sqrt{3}}{6}$

Step 2.7.14.2

Factor $6$ out of $6 \sqrt{3}$ .

$\frac{20}{x} = \frac{6 \cdot 2 + 6 (\sqrt{3})}{6}$

Step 2.7.14.3

Factor $6$ out of $6 (2) + 6 (\sqrt{3})$ .

$\frac{20}{x} = \frac{6 (2 + \sqrt{3})}{6}$

Step 2.7.14.4

Cancel the common factors.

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Step 2.7.14.4.1

Factor $6$ out of $6$ .

$\frac{20}{x} = \frac{6 (2 + \sqrt{3})}{6 (1)}$

Step 2.7.14.4.2

Cancel the common factor.

$\frac{20}{x} = \frac{6 (2 + \sqrt{3})}{6 \cdot 1}$

Step 2.7.14.4.3

Rewrite the expression.

$\frac{20}{x} = \frac{2 + \sqrt{3}}{1}$

Step 2.7.14.4.4

Divide $2 + \sqrt{3}$ by $1$ .

$\frac{20}{x} = 2 + \sqrt{3}$

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.

$x, 1, 1$

Step 3.2

The LCM of one and any expression is the expression.

$x$

Step 4

Multiply each term in $\frac{20}{x} = 2 + \sqrt{3}$ by $x$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{20}{x} = 2 + \sqrt{3}$ by $x$ .

$\frac{20}{x} x = 2 x + \sqrt{3} x$

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.

$\frac{20}{x} x = 2 x + \sqrt{3} x$

Step 4.2.1.2

Rewrite the expression.

$20 = 2 x + \sqrt{3} x$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $2 x + \sqrt{3} x = 20$ .

$2 x + \sqrt{3} x = 20$

Step 5.2

Factor $x$ out of $2 x + \sqrt{3} x$ .

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Step 5.2.1

Factor $x$ out of $2 x$ .

$x \cdot 2 + \sqrt{3} x = 20$

Step 5.2.2

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

$x \cdot 2 + x \sqrt{3} = 20$

Step 5.2.3

Factor $x$ out of $x \cdot 2 + x \sqrt{3}$ .

$x (2 + \sqrt{3}) = 20$

Step 5.3

Divide each term in $x (2 + \sqrt{3}) = 20$ by $2 + \sqrt{3}$ and simplify.

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Step 5.3.1

Divide each term in $x (2 + \sqrt{3}) = 20$ by $2 + \sqrt{3}$ .

$\frac{x (2 + \sqrt{3})}{2 + \sqrt{3}} = \frac{20}{2 + \sqrt{3}}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2 + \sqrt{3}$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{x (2 + \sqrt{3})}{2 + \sqrt{3}} = \frac{20}{2 + \sqrt{3}}$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{20}{2 + \sqrt{3}}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply $\frac{20}{2 + \sqrt{3}}$ by $\frac{2 - \sqrt{3}}{2 - \sqrt{3}}$ .

$x = \frac{20}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}}$

Step 5.3.3.2

Multiply $\frac{20}{2 + \sqrt{3}}$ by $\frac{2 - \sqrt{3}}{2 - \sqrt{3}}$ .

$x = \frac{20 (2 - \sqrt{3})}{(2 + \sqrt{3}) (2 - \sqrt{3})}$

Step 5.3.3.3

Expand the denominator using the FOIL method.

$x = \frac{20 (2 - \sqrt{3})}{4 - 2 \sqrt{3} + \sqrt{3} \cdot 2 - {\sqrt{3}}^{2}}$

Step 5.3.3.4

Simplify.

$x = \frac{20 (2 - \sqrt{3})}{1}$

Step 5.3.3.5

Divide $20 (2 - \sqrt{3})$ by $1$ .

$x = 20 (2 - \sqrt{3})$

Step 5.3.3.6

Apply the distributive property.

$x = 20 \cdot 2 + 20 (- \sqrt{3})$

Step 5.3.3.7

Multiply.

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Step 5.3.3.7.1

Multiply $20$ by $2$ .

$x = 40 + 20 (- \sqrt{3})$

Step 5.3.3.7.2

Multiply $- 1$ by $20$ .

$x = 40 - 20 \sqrt{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = 40 - 20 \sqrt{3}$

Decimal Form:

$x = 5.35898384 \dots$