Find the Cosecant Given the Point (-( square root of 5)/3,-2/3)

$(- \frac{\sqrt{5}}{3}, - \frac{2}{3})$

To find the $\csc (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{\sqrt{5}}{3}, - \frac{2}{3})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{\sqrt{5}}{3}, 0)$ , and $(- \frac{\sqrt{5}}{3}, - \frac{2}{3})$ .

Opposite : $- \frac{2}{3}$

Adjacent : $- \frac{\sqrt{5}}{3}$

Find the hypotenuse using Pythagorean theorem $c = \sqrt{a^{2} + b^{2}}$ .

Tap for more steps...

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

Tap for more steps...

Apply the product rule to $- \frac{\sqrt{5}}{3}$ .

$\sqrt{{(- 1)}^{2} {(\frac{\sqrt{5}}{3})}^{2} + {(- \frac{2}{3})}^{2}}$

Apply the product rule to $\frac{\sqrt{5}}{3}$ .

$\sqrt{{(- 1)}^{2} \frac{{\sqrt{5}}^{2}}{3^{2}} + {(- \frac{2}{3})}^{2}}$

Simplify the expression.

Tap for more steps...

Raise $- 1$ to the power of $2$ .

$\sqrt{1 \frac{{\sqrt{5}}^{2}}{3^{2}} + {(- \frac{2}{3})}^{2}}$

Multiply $\frac{{\sqrt{5}}^{2}}{3^{2}}$ by $1$ .

$\sqrt{\frac{{\sqrt{5}}^{2}}{3^{2}} + {(- \frac{2}{3})}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Tap for more steps...

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sqrt{\frac{{(5^{\frac{1}{2}})}^{2}}{3^{2}} + {(- \frac{2}{3})}^{2}}$

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

$\sqrt{\frac{5^{\frac{1}{2} \cdot 2}}{3^{2}} + {(- \frac{2}{3})}^{2}}$

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

$\sqrt{\frac{5^{\frac{2}{2}}}{3^{2}} + {(- \frac{2}{3})}^{2}}$

Cancel the common factor of $2$ .

Tap for more steps...

Cancel the common factor.

$\sqrt{\frac{5^{\frac{2}{2}}}{3^{2}} + {(- \frac{2}{3})}^{2}}$

Rewrite the expression.

$\sqrt{\frac{5^{1}}{3^{2}} + {(- \frac{2}{3})}^{2}}$

Evaluate the exponent.

$\sqrt{\frac{5}{3^{2}} + {(- \frac{2}{3})}^{2}}$

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

$\sqrt{\frac{5}{9} + {(- \frac{2}{3})}^{2}}$

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

Tap for more steps...

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

$\sqrt{\frac{5}{9} + {(- 1)}^{2} {(\frac{2}{3})}^{2}}$

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

$\sqrt{\frac{5}{9} + {(- 1)}^{2} \frac{2^{2}}{3^{2}}}$

Raise $- 1$ to the power of $2$ .

$\sqrt{\frac{5}{9} + 1 \frac{2^{2}}{3^{2}}}$

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

$\sqrt{\frac{5}{9} + \frac{2^{2}}{3^{2}}}$

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

$\sqrt{\frac{5}{9} + \frac{4}{3^{2}}}$

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

$\sqrt{\frac{5}{9} + \frac{4}{9}}$

Combine the numerators over the common denominator.

$\sqrt{\frac{5 + 4}{9}}$

Add $5$ and $4$ .

$\sqrt{\frac{9}{9}}$

Divide $9$ by $9$ .

$\sqrt{1}$

Any root of $1$ is $1$ .

$1$

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore $\csc (θ) = \frac{1}{- \frac{2}{3}}$ .

$\frac{1}{- \frac{2}{3}}$

Simplify $\csc (θ)$ .

Tap for more steps...

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

Tap for more steps...

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

$\csc (θ) = \frac{- 1 \cdot - 1}{- \frac{2}{3}}$

Move the negative in front of the fraction.

$\csc (θ) = - \frac{1}{\frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$\csc (θ) = - (1 (\frac{3}{2}))$

Multiply $\frac{3}{2}$ by $1$ .

$\csc (θ) = - \frac{3}{2}$

Approximate the result.

$\csc (θ) = - \frac{3}{2} \approx - 1.5$

Do you know how to Find the Cosecant Given the Point (-( square root of 5)/3,-2/3)? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.