Evaluate tan(arccos(-( square root of 3)/2))

$\tan (\arccos (- \frac{\sqrt{3}}{2}))$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Add $1$ and $1$ .

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

Step 6

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 7

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.57735026 \dots$