Find the Cosecant Given the Point (1,- square root of 3)

$(1, - \sqrt{3})$

To find the $\csc (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(1, - \sqrt{3})$ , draw the triangle between the three points $(0, 0)$ , $(1, 0)$ , and $(1, - \sqrt{3})$ .

Opposite : $- \sqrt{3}$

Adjacent : $1$

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

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Simplify the expression.

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One to any power is one.

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

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

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

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

$\sqrt{1 + 1 {\sqrt{3}}^{2}}$

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

$\sqrt{1 + {\sqrt{3}}^{2}}$

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$\sqrt{1 + 3}$

Simplify the expression.

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Add $1$ and $3$ .

$\sqrt{4}$

Rewrite $4$ as $2^{2}$ .

$\sqrt{2^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$2$

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

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

Simplify $\csc (θ)$ .

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Move the negative in front of the fraction.

$\csc (θ) = - \frac{2}{\sqrt{3}}$

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

$\csc (θ) = - \frac{2 \sqrt{3}}{3}$

Evaluate the exponent.

$\csc (θ) = - \frac{2 \sqrt{3}}{3}$

Approximate the result.

$\csc (θ) = - \frac{2 \sqrt{3}}{3} \approx - 1.15470053$

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