Trigonometry Examples

$\frac{\tan (\frac{π}{3}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1

Use a sum or difference formula on the numerator.

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Step 1.1

The angle $\frac{π}{3}$ is an angle where the values of the six trigonometric functions are known. Because this is the case, add $0$ to keep the value the same.

$\frac{(\tan (\frac{π}{3} + 0)) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.2

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

$\frac{(\frac{\tan (\frac{π}{3}) + \tan (0)}{1 - \tan (\frac{π}{3}) \tan (0)}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.3

Simplify the numerator.

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Step 1.3.1

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

$\frac{(\frac{\sqrt{3} + \tan (0)}{1 - \tan (\frac{π}{3}) \tan (0)}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.3.2

The exact value of $\tan (0)$ is $0$ .

$\frac{(\frac{\sqrt{3} + 0}{1 - \tan (\frac{π}{3}) \tan (0)}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.3.3

Add $\sqrt{3}$ and $0$ .

$\frac{(\frac{\sqrt{3}}{1 - \tan (\frac{π}{3}) \tan (0)}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.4

Simplify the denominator.

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Step 1.4.1

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

$\frac{(\frac{\sqrt{3}}{1 - \sqrt{3} \tan (0)}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.4.2

The exact value of $\tan (0)$ is $0$ .

$\frac{(\frac{\sqrt{3}}{1 - \sqrt{3} \cdot 0}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.4.3

Multiply $- \sqrt{3} \cdot 0$ .

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Step 1.4.3.1

Multiply $0$ by $- 1$ .

$\frac{(\frac{\sqrt{3}}{1 + 0 \sqrt{3}}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.4.3.2

Multiply $0$ by $\sqrt{3}$ .

$\frac{(\frac{\sqrt{3}}{1 + 0}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.4.4

Add $1$ and $0$ .

$\frac{(\frac{\sqrt{3}}{1}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.5

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

$\frac{(\sqrt{3}) + \tan (\frac{π}{4})}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.6

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{\sqrt{3} + 1}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.7

The angle $\frac{π}{4}$ is an angle where the values of the six trigonometric functions are known. Because this is the case, add $0$ to keep the value the same.

$\frac{\sqrt{3} + \tan (\frac{π}{4} + 0)}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.8

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

$\frac{\frac{\sqrt{3} (\tan (\frac{π}{4}) + \tan (0))}{1 - \tan (\frac{π}{4}) \tan (0)}}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.9

Simplify the numerator.

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Step 1.9.1

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{\sqrt{3} + \frac{1 + \tan (0)}{1 - \tan (\frac{π}{4}) \tan (0)}}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.9.2

The exact value of $\tan (0)$ is $0$ .

$\frac{\sqrt{3} + \frac{1 + 0}{1 - \tan (\frac{π}{4}) \tan (0)}}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.9.3

Add $1$ and $0$ .

$\frac{\sqrt{3} + \frac{1}{1 - \tan (\frac{π}{4}) \tan (0)}}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.10

Simplify the denominator.

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Step 1.10.1

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$\frac{\sqrt{3} + \frac{1}{1 - 1 \cdot (1 \tan (0))}}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.10.2

Multiply $- 1$ by $1$ .

$\frac{\sqrt{3} + \frac{1}{1 - 1 \tan (0)}}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.10.3

The exact value of $\tan (0)$ is $0$ .

$\frac{\sqrt{3} + \frac{1}{1 - 1 \cdot 0}}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.10.4

Multiply $- 1$ by $0$ .

$\frac{\sqrt{3} + \frac{1}{1 + 0}}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.10.5

Add $1$ and $0$ .

$\frac{\sqrt{3} + \frac{1}{1}}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.11

Cancel the common factor of $1$ .

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Step 1.11.1

Cancel the common factor.

$\frac{\sqrt{3} + \frac{1}{1}}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 1.11.2

Rewrite the expression.

$\frac{\sqrt{3} + 1}{1 - \tan (\frac{π}{3}) \tan (\frac{π}{4})}$

Step 2

Simplify.

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Step 2.1

Simplify the denominator.

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Step 2.1.1

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

$\frac{\sqrt{3} + 1}{1 - \sqrt{3} \tan (\frac{π}{4})}$

Step 2.1.2

The exact value of $\tan (\frac{π}{4})$ is $1$ .

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

Step 2.1.3

Multiply $- 1$ by $1$ .

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

Step 2.2

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

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

Step 2.3

Combine fractions.

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Step 2.3.1

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

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

Step 2.3.2

Expand the denominator using the FOIL method.

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

Step 2.3.3

Simplify.

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

Step 2.4

Simplify the numerator.

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Step 2.4.1

Reorder terms.

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

Step 2.4.2

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

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

Step 2.4.3

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

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

Step 2.4.4

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

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

Step 2.4.5

Add $1$ and $1$ .

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

Step 2.5

Rewrite ${(1 + \sqrt{3})}^{2}$ as $(1 + \sqrt{3}) (1 + \sqrt{3})$ .

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

Step 2.6

Expand $(1 + \sqrt{3}) (1 + \sqrt{3})$ using the FOIL Method.

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Step 2.6.1

Apply the distributive property.

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

Step 2.6.2

Apply the distributive property.

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

Step 2.6.3

Apply the distributive property.

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

Step 2.7

Simplify and combine like terms.

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

Multiply $1$ by $1$ .

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

Step 2.7.1.2

Multiply $\sqrt{3}$ by $1$ .

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

Step 2.7.1.3

Multiply $\sqrt{3}$ by $1$ .

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

Step 2.7.1.4

Combine using the product rule for radicals.

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

Step 2.7.1.5

Multiply $3$ by $3$ .

$\frac{1 + \sqrt{3} + \sqrt{3} + \sqrt{9}}{- 2}$

Step 2.7.1.6

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

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

Step 2.7.1.7

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

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

Step 2.7.2

Add $1$ and $3$ .

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

Step 2.7.3

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

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

Step 2.8

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

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Step 2.8.1

Factor $2$ out of $4$ .

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

Step 2.8.2

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

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

Step 2.8.3

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

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

Step 2.8.4

Move the negative one from the denominator of $\frac{2 + \sqrt{3}}{- 1}$ .

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

Step 2.9

Rewrite $- 1 \cdot (2 + \sqrt{3})$ as $- (2 + \sqrt{3})$ .

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

Step 2.10

Apply the distributive property.

$- 1 \cdot 2 - \sqrt{3}$

Step 2.11

Multiply $- 1$ by $2$ .

$- 2 - \sqrt{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 2 - \sqrt{3}$

Decimal Form:

$- 3.73205080 \dots$