Find the Cosecant Given the Point (-6,6)

$(- 6, 6)$

To find the $\csc (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- 6, 6)$ , draw the triangle between the three points $(0, 0)$ , $(- 6, 0)$ , and $(- 6, 6)$ .

Opposite : $6$

Adjacent : $- 6$

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

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Raise $- 6$ to the power of $2$ .

$\sqrt{36 + {(6)}^{2}}$

Raise $6$ to the power of $2$ .

$\sqrt{36 + 36}$

Add $36$ and $36$ .

$\sqrt{72}$

Rewrite $72$ as $6^{2} \cdot 2$ .

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Factor $36$ out of $72$ .

$\sqrt{36 (2)}$

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

$\sqrt{6^{2} \cdot 2}$

Pull terms out from under the radical.

$6 \sqrt{2}$

$\csc (θ) = \frac{Hypotenuse}{Opposite}$ therefore $\csc (θ) = \frac{6 \sqrt{2}}{6}$ .

$\frac{6 \sqrt{2}}{6}$

Cancel the common factor of $6$ .

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Cancel the common factor.

$\csc (θ) = \frac{6 \sqrt{2}}{6}$

Divide $\sqrt{2}$ by $1$ .

$\csc (θ) = \sqrt{2}$

Approximate the result.

$\csc (θ) = \sqrt{2} \approx 1.41421356$

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