Trigonometry Examples

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

Step 1

Cancel the common factor of $1$ and $- 1$ .

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Step 1.1

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

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

Step 1.2

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

Combine and simplify the denominator.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

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

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Step 3.6.1

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

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

Step 3.6.2

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

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

Step 3.6.3

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

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

Step 3.6.4

Cancel the common factor of $2$ .

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Step 3.6.4.1

Cancel the common factor.

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

Step 3.6.4.2

Rewrite the expression.

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

Step 3.6.5

Evaluate the exponent.

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

Step 4

The exact value of $\arctan (- \frac{\sqrt{3}}{3})$ is $- \frac{π}{6}$ .

$- \frac{π}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$- \frac{π}{6}$

Decimal Form:

$- 0.52359877 \dots$