Find the Exact Value arccsc(csc((4pi)/3))

$arccsc (\csc (\frac{4 π}{3}))$

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosecant is negative in the third quadrant.

$arccsc (- \csc (\frac{π}{3}))$

The exact value of $\csc (\frac{π}{3})$ is $\frac{2}{\sqrt{3}}$ .

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

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 1.04719755 \dots$

Find the Exact Value arccsc(csc((4pi)/3))

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