Trigonometry Examples

$\tan (75)$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $75$ can be split into $30 + 45$ .

$\tan (30 + 45)$

Step 2

Use the sum formula for tangent to simplify the expression. The formula states that $\tan (A + B) = \frac{\tan (A) + \tan (B)}{1 - \tan (A) \tan (B)}$ .

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply $- 1$ by $1$ .

$\frac{\frac{\sqrt{3} + 3}{3}}{1 - \frac{\sqrt{3}}{3}}$

Step 4.4

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

$\frac{\frac{\sqrt{3} + 3}{3}}{\frac{3}{3} - \frac{\sqrt{3}}{3}}$

Step 4.5

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{3} + 3}{3}}{\frac{3 - \sqrt{3}}{3}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{3} + 3}{3} \cdot \frac{3}{3 - \sqrt{3}}$

Step 6

Cancel the common factor of $3$ .

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Step 6.1

Cancel the common factor.

$\frac{\sqrt{3} + 3}{3} \cdot \frac{3}{3 - \sqrt{3}}$

Step 6.2

Rewrite the expression.

$(\sqrt{3} + 3) \frac{1}{3 - \sqrt{3}}$

Step 7

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

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

Step 8

Simplify terms.

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Step 8.1

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

$(\sqrt{3} + 3) \frac{3 + \sqrt{3}}{(3 - \sqrt{3}) (3 + \sqrt{3})}$

Step 8.2

Expand the denominator using the FOIL method.

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

Step 8.3

Simplify.

$(\sqrt{3} + 3) \frac{3 + \sqrt{3}}{6}$

Step 8.4

Apply the distributive property.

$\sqrt{3} \frac{3 + \sqrt{3}}{6} + 3 \frac{3 + \sqrt{3}}{6}$

Step 8.5

Combine $\sqrt{3}$ and $\frac{3 + \sqrt{3}}{6}$ .

$\frac{\sqrt{3} (3 + \sqrt{3})}{6} + 3 \frac{3 + \sqrt{3}}{6}$

Step 8.6

Cancel the common factor of $3$ .

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Step 8.6.1

Factor $3$ out of $6$ .

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

Step 8.6.2

Cancel the common factor.

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

Step 8.6.3

Rewrite the expression.

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

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

$\frac{\sqrt{3} \cdot 3 + \sqrt{3} \sqrt{3}}{6} + \frac{3 + \sqrt{3}}{2}$

Step 9.2

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

$\frac{3 \cdot \sqrt{3} + \sqrt{3} \sqrt{3}}{6} + \frac{3 + \sqrt{3}}{2}$

Step 9.3

Combine using the product rule for radicals.

$\frac{3 \cdot \sqrt{3} + \sqrt{3 \cdot 3}}{6} + \frac{3 + \sqrt{3}}{2}$

Step 9.4

Simplify each term.

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Step 9.4.1

Multiply $3$ by $3$ .

$\frac{3 \sqrt{3} + \sqrt{9}}{6} + \frac{3 + \sqrt{3}}{2}$

Step 9.4.2

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

$\frac{3 \sqrt{3} + \sqrt{3^{2}}}{6} + \frac{3 + \sqrt{3}}{2}$

Step 9.4.3

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

$\frac{3 \sqrt{3} + 3}{6} + \frac{3 + \sqrt{3}}{2}$

Step 9.5

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

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Step 9.5.1

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

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

Step 9.5.2

Factor $3$ out of $3$ .

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

Step 9.5.3

Factor $3$ out of $3 (\sqrt{3}) + 3 (1)$ .

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

Step 9.5.4

Cancel the common factors.

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Step 9.5.4.1

Factor $3$ out of $6$ .

$\frac{3 (\sqrt{3} + 1)}{3 (2)} + \frac{3 + \sqrt{3}}{2}$

Step 9.5.4.2

Cancel the common factor.

$\frac{3 (\sqrt{3} + 1)}{3 \cdot 2} + \frac{3 + \sqrt{3}}{2}$

Step 9.5.4.3

Rewrite the expression.

$\frac{\sqrt{3} + 1}{2} + \frac{3 + \sqrt{3}}{2}$

Step 10

Simplify terms.

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Step 10.1

Combine the numerators over the common denominator.

$\frac{\sqrt{3} + 1 + 3 + \sqrt{3}}{2}$

Step 10.2

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

$\frac{1 + 3 + 2 \sqrt{3}}{2}$

Step 10.3

Add $1$ and $3$ .

$\frac{4 + 2 \sqrt{3}}{2}$

Step 10.4

Cancel the common factor of $4 + 2 \sqrt{3}$ and $2$ .

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Step 10.4.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 2 + 2 \sqrt{3}}{2}$

Step 10.4.2

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

$\frac{2 \cdot 2 + 2 (\sqrt{3})}{2}$

Step 10.4.3

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

$\frac{2 (2 + \sqrt{3})}{2}$

Step 10.4.4

Cancel the common factors.

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Step 10.4.4.1

Factor $2$ out of $2$ .

$\frac{2 (2 + \sqrt{3})}{2 (1)}$

Step 10.4.4.2

Cancel the common factor.

$\frac{2 (2 + \sqrt{3})}{2 \cdot 1}$

Step 10.4.4.3

Rewrite the expression.

$\frac{2 + \sqrt{3}}{1}$

Step 10.4.4.4

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

$2 + \sqrt{3}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$2 + \sqrt{3}$

Decimal Form:

$3.73205080 \dots$