Trigonometry Examples

arctan (- \frac{1}{\sqrt{3}})

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

Step 1

To convert radians to degrees, multiply by

\frac{180}{π}

$\frac{180}{π}$ , since a full circle is

360 °

$360 °$ or

2 π

$2 π$ radians.

(arctan (- \frac{1}{\sqrt{3}})) \cdot \frac{180 °}{π}

$(\arctan (- \frac{1}{\sqrt{3}})) \cdot \frac{180 °}{π}$

Step 2

Multiply

\frac{1}{\sqrt{3}}

$\frac{1}{\sqrt{3}}$ by

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

arctan (- (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})) \cdot \frac{180}{π}

$\arctan (- (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})) \cdot \frac{180}{π}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply

\frac{1}{\sqrt{3}}

$\frac{1}{\sqrt{3}}$ by

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

arctan (- \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}) \cdot \frac{180}{π}

$\arctan (- \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}) \cdot \frac{180}{π}$

Step 3.2

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

arctan (- \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}) \cdot \frac{180}{π}

$\arctan (- \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}) \cdot \frac{180}{π}$

Step 3.3

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

arctan (- \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}) \cdot \frac{180}{π}

$\arctan (- \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}) \cdot \frac{180}{π}$

Step 3.4

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

arctan (- \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}) \cdot \frac{180}{π}

$\arctan (- \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}) \cdot \frac{180}{π}$

Step 3.5

Add

1

$1$ and

1

$1$ .

arctan (- \frac{\sqrt{3}}{{\sqrt{3}}^{2}}) \cdot \frac{180}{π}

$\arctan (- \frac{\sqrt{3}}{{\sqrt{3}}^{2}}) \cdot \frac{180}{π}$

Step 3.6

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 3.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

arctan ⎛ ⎜ ⎜ ⎝ - \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} ⎞ ⎟ ⎟ ⎠ \cdot \frac{180}{π}

$\arctan (- \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}) \cdot \frac{180}{π}$

Step 3.6.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

arctan (- \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}) \cdot \frac{180}{π}

$\arctan (- \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}) \cdot \frac{180}{π}$

Step 3.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

arctan (- \frac{\sqrt{3}}{3^{\frac{2}{2}}}) \cdot \frac{180}{π}

$\arctan (- \frac{\sqrt{3}}{3^{\frac{2}{2}}}) \cdot \frac{180}{π}$

Step 3.6.4

Cancel the common factor of

2

$2$ .

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Step 3.6.4.1

Cancel the common factor.

$\arctan (- \frac{\sqrt{3}}{3^{\frac{2}{2}}}) \cdot \frac{180}{π}$

Step 3.6.4.2

Rewrite the expression.

$\arctan (- \frac{\sqrt{3}}{3^{1}}) \cdot \frac{180}{π}$

Step 3.6.5

Evaluate the exponent.

$\arctan (- \frac{\sqrt{3}}{3}) \cdot \frac{180}{π}$

Step 4

The exact value of

$\arctan (- \frac{\sqrt{3}}{3})$ is

$- \frac{π}{6}$ .

$- \frac{π}{6} \cdot \frac{180}{π}$

Step 5

Cancel the common factor of

$π$ .

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Step 5.1

Move the leading negative in

$- \frac{π}{6}$ into the numerator.

$\frac{- π}{6} \cdot \frac{180}{π}$

Step 5.2

Factor

$π$ out of

$- π$ .

$\frac{π \cdot - 1}{6} \cdot \frac{180}{π}$

Step 5.3

Cancel the common factor.

$\frac{π \cdot - 1}{6} \cdot \frac{180}{π}$

Step 5.4

Rewrite the expression.

$\frac{- 1}{6} \cdot 180$

Step 6

Cancel the common factor of

$6$ .

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Step 6.1

Factor

$6$ out of

$180$ .

$\frac{- 1}{6} \cdot (6 (30))$

Step 6.2

Cancel the common factor.

$\frac{- 1}{6} \cdot (6 \cdot 30)$

Step 6.3

Rewrite the expression.

$- 1 \cdot 30$

Step 7

Multiply

$- 1$ by

$30$ .

$- 30$

Step 8

Convert to a decimal.

$- 30 °$