Trigonometry Examples

$\tan (15)$

Step 1

First, split the angle into two angles where the values of the six trigonometric functions are known. In this case, $15$ can be split into $60 - 45$ .

$\tan (60 - 45)$

Step 2

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

$\frac{\tan (60) - \tan (45)}{1 + \tan (60) \tan (45)}$

Step 3

Remove parentheses.

$\frac{\tan (60) - \tan (45)}{1 + \tan (60) \tan (45)}$

Step 4

Simplify the numerator.

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Step 4.1

The exact value of $\tan (60)$ is $\sqrt{3}$ .

$\frac{\sqrt{3} - \tan (45)}{1 + \tan (60) \tan (45)}$

Step 4.2

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

$\frac{\sqrt{3} - 1 \cdot 1}{1 + \tan (60) \tan (45)}$

Step 4.3

Multiply $- 1$ by $1$ .

$\frac{\sqrt{3} - 1}{1 + \tan (60) \tan (45)}$

Step 5

Simplify the denominator.

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Step 5.1

The exact value of $\tan (60)$ is $\sqrt{3}$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

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 7

Combine fractions.

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Step 7.1

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 7.2

Expand the denominator using the FOIL method.

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

Step 7.3

Simplify.

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

Step 8

Simplify the numerator.

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Step 8.1

Factor $- 1$ out of $\sqrt{3}$ .

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

Step 8.2

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 8.3

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

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

Step 8.4

Reorder terms.

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

Step 8.5

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

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

Step 8.6

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

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

Step 8.7

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

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

Step 8.8

Add $1$ and $1$ .

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

Step 9

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

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

Step 10

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

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Step 10.1

Apply the distributive property.

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

Step 10.2

Apply the distributive property.

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

Step 10.3

Apply the distributive property.

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

Step 11

Simplify and combine like terms.

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Step 11.1

Simplify each term.

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Step 11.1.1

Multiply $1$ by $1$ .

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

Step 11.1.2

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

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

Step 11.1.3

Multiply $- 1$ by $1$ .

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

Step 11.1.4

Multiply $- \sqrt{3} (- \sqrt{3})$ .

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Step 11.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 11.1.4.2

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

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

Step 11.1.4.3

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

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

Step 11.1.4.4

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

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

Step 11.1.4.5

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

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

Step 11.1.4.6

Add $1$ and $1$ .

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

Step 11.1.5

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

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Step 11.1.5.1

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

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

Step 11.1.5.2

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

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

Step 11.1.5.3

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

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

Step 11.1.5.4

Cancel the common factor of $2$ .

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Step 11.1.5.4.1

Cancel the common factor.

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

Step 11.1.5.4.2

Rewrite the expression.

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

Step 11.1.5.5

Evaluate the exponent.

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

Step 11.2

Add $1$ and $3$ .

$\frac{- 1 (4 - \sqrt{3} - \sqrt{3})}{- 2}$

Step 11.3

Subtract $\sqrt{3}$ from $- \sqrt{3}$ .

$\frac{- 1 (4 - 2 \sqrt{3})}{- 2}$

Step 12

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

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Step 12.1

Factor $2$ out of $- 1 (4 - 2 \sqrt{3})$ .

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

Step 12.2

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

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

Step 13

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

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

Step 14

Apply the distributive property.

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

Step 15

Multiply $- 1$ by $2$ .

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

Step 16

Multiply $- 1 (- \sqrt{3})$ .

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Step 16.1

Multiply $- 1$ by $- 1$ .

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

Step 16.2

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

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

Step 17

Apply the distributive property.

$- - 2 - \sqrt{3}$

Step 18

Multiply $- 1$ by $- 2$ .

$2 - \sqrt{3}$

Step 19

The result can be shown in multiple forms.

Exact Form:

$2 - \sqrt{3}$

Decimal Form:

$0.26794919 \dots$