How do you use special angles to find the ratios of csc 315 degrees?

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How do you use special angles to find the ratios of csc 315 degrees?

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Answer 1 · 2015-10-17T17:59:28

$315^{\circ} = 45^{\circ}$ $315^@ = 45^@$ less than a full circle
$XXX$ $color(white)("XXX")$ i.e. $315^{\circ} \equiv - 45^{\circ}$ $315^@-= -45^@$
$csc (- 45^{\circ}) = \frac{1}{sin (- 45^{\circ})} = - \frac{1}{sin (45^{\circ})} = - \sqrt{2}$ $csc(-45^@) =1/sin(-45^@)=-1/sin(45^@)= -sqrt(2)$

Explanation:

Since $(- 45^{\circ})$ $(-45^@)$ is in Quadrant IV, $sin$ $sin$ (and therefore $csc$ $csc$ ) is negative.

The $45^{\circ}$ $45^@$ angle is a no-right angle in a right-angled equilateral triangle. If the equal length arms of the triangle are of length 1, then the hypotenuse is of length $\sqrt{2}$ $sqrt(2)$ (Pythagorean Theorem).

$sin = \frac{opposite}{hypotenuse}$ $sin = ("opposite")/("hypotenuse")$
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How do you use special angles to find the ratios of csc 315 degrees?

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