Evaluate tan(165 degrees )

$\tan (165 °)$

Step 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.

$- \tan (15)$

Step 2

Split $15$ into two angles where the values of the six trigonometric functions are known.

$- \tan (45 - 30)$

Step 3

Separate negation.

$- \tan (45 - (30))$

Step 4

Apply the difference of angles identity.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

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

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

Combine.

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

Apply the distributive property.

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

Cancel the common factor of $3$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $3$ by $1$ .

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

Simplify the denominator.

Multiply $3$ by $1$ .

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

Cancel the common factor of $3$ .

Factor $3$ out of $3 \cdot 1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Expand the denominator using the FOIL method.

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

Simplify.

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

Simplify the numerator.

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

$- \frac{{(3 - \sqrt{3})}^{1} (3 - \sqrt{3})}{6}$

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

$- \frac{{(3 - \sqrt{3})}^{1} {(3 - \sqrt{3})}^{1}}{6}$

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

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

Add $1$ and $1$ .

$- \frac{{(3 - \sqrt{3})}^{2}}{6}$

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$- \frac{9 + 3 (- \sqrt{3}) - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Multiply $- 1$ by $3$ .

$- \frac{9 - 3 \sqrt{3} - \sqrt{3} \cdot 3 - \sqrt{3} (- \sqrt{3})}{6}$

Multiply $3$ by $- 1$ .

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} - \sqrt{3} (- \sqrt{3})}{6}$

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

Multiply $- 1$ by $- 1$ .

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 1 \sqrt{3} \sqrt{3}}{6}$

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

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

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

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} \sqrt{3}}{6}$

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

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1} {\sqrt{3}}^{1}}{6}$

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

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {\sqrt{3}}^{1 + 1}}{6}$

Add $1$ and $1$ .

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

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

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

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + {(3^{\frac{1}{2}})}^{2}}{6}$

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

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{\frac{1}{2} \cdot 2}}{6}$

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$- \frac{9 - 3 \sqrt{3} - 3 \sqrt{3} + 3^{1}}{6}$

Evaluate the exponent.

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

Add $9$ and $3$ .

$- \frac{12 - 3 \sqrt{3} - 3 \sqrt{3}}{6}$

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

$- \frac{12 - 6 \sqrt{3}}{6}$

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

Factor $6$ out of $12$ .

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

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

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

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

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

Cancel the common factors.

Factor $6$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

$- 2 - - \sqrt{3}$

Multiply $- - \sqrt{3}$ .

Multiply $- 1$ by $- 1$ .

$- 2 + 1 \sqrt{3}$

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

$- 2 + \sqrt{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- 2 + \sqrt{3}$

Decimal Form:

$- 0.26794919 \dots$

Evaluate tan(165 degrees )

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