Find the Cosecant Given the Point (2 square root of 5,4)

$(2 \sqrt{5}, 4)$

To find the $\csc (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(2 \sqrt{5}, 4)$ , draw the triangle between the three points $(0, 0)$ , $(2 \sqrt{5}, 0)$ , and $(2 \sqrt{5}, 4)$ .

Opposite : $4$

Adjacent : $2 \sqrt{5}$

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

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Simplify the expression.

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Apply the product rule to $2 \sqrt{5}$ .

$\sqrt{2^{2} {\sqrt{5}}^{2} + {(4)}^{2}}$

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

$\sqrt{4 {\sqrt{5}}^{2} + {(4)}^{2}}$

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

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

$\sqrt{4 {(5^{\frac{1}{2}})}^{2} + {(4)}^{2}}$

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

$\sqrt{4 \cdot 5^{\frac{1}{2} \cdot 2} + {(4)}^{2}}$

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

$\sqrt{4 \cdot 5^{\frac{2}{2}} + {(4)}^{2}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\sqrt{4 \cdot 5^{\frac{2}{2}} + {(4)}^{2}}$

Rewrite the expression.

$\sqrt{4 \cdot 5^{1} + {(4)}^{2}}$

Evaluate the exponent.

$\sqrt{4 \cdot 5 + {(4)}^{2}}$

Simplify the expression.

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Multiply $4$ by $5$ .

$\sqrt{20 + {(4)}^{2}}$

Raise $4$ to the power of $2$ .

$\sqrt{20 + 16}$

Add $20$ and $16$ .

$\sqrt{36}$

Rewrite $36$ as $6^{2}$ .

$\sqrt{6^{2}}$

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

$6$

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore $\csc (θ) = \frac{6}{4}$ .

$\frac{6}{4}$

Cancel the common factor of $6$ and $4$ .

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Factor $2$ out of $6$ .

$\csc (θ) = \frac{2 (3)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

$\csc (θ) = \frac{3}{2}$

Approximate the result.

$\csc (θ) = \frac{3}{2} \approx 1.5$

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