Evaluate -csc((5pi)/3)^2

$- \csc^{2} (\frac{5 π}{3})$

Step 1

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 fourth quadrant.

$- {(- \csc (\frac{π}{3}))}^{2}$

Step 2

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

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{2 \sqrt{3}}{3}$ .

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

Apply the product rule to $\frac{2 \sqrt{3}}{3}$ .

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

Apply the product rule to $2 \sqrt{3}$ .

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

Step 6

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

Move ${(- 1)}^{2}$ .

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

Multiply ${(- 1)}^{2}$ by $- 1$ .

Raise $- 1$ to the power of $1$ .

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

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

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

Add $2$ and $1$ .

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

Step 7

Raise $- 1$ to the power of $3$ .

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

Step 8

Simplify the numerator.

Raise $2$ to the power of $2$ .

$- \frac{4 {\sqrt{3}}^{2}}{3^{2}}$

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

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

$- \frac{4 {(3^{\frac{1}{2}})}^{2}}{3^{2}}$

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

$- \frac{4 \cdot 3^{\frac{1}{2} \cdot 2}}{3^{2}}$

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

$- \frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$- \frac{4 \cdot 3^{\frac{2}{2}}}{3^{2}}$

Rewrite the expression.

$- \frac{4 \cdot 3^{1}}{3^{2}}$

Evaluate the exponent.

$- \frac{4 \cdot 3}{3^{2}}$

Step 9

Reduce the expression by cancelling the common factors.

Raise $3$ to the power of $2$ .

$- \frac{4 \cdot 3}{9}$

Multiply $4$ by $3$ .

$- \frac{12}{9}$

Cancel the common factor of $12$ and $9$ .

Factor $3$ out of $12$ .

$- \frac{3 (4)}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$- \frac{3 \cdot 4}{3 \cdot 3}$

Cancel the common factor.

$- \frac{3 \cdot 4}{3 \cdot 3}$

Rewrite the expression.

$- \frac{4}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{4}{3}$

Decimal Form:

$- 1.\overline{3}$

Mixed Number Form:

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

Evaluate -csc((5pi)/3)^2

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