Evaluate each expression.
$\(\csc\)225\(\degree\)$

• A. $1$
• B. $-\(\frac{\sqrt2}{2}\)$
• C. $\(\sqrt\)2$
• D. $-\(\sqrt\)2$

Answer: (D)

Verified step by step guidance

1. Understand that the cosecant function, \( \csc \theta \), is the reciprocal of the sine function, \( \sin \theta \). Therefore, \( \csc 225^\circ = \frac{1}{\sin 225^\circ} \).
2. Recognize that 225° is in the third quadrant of the unit circle, where both sine and cosine are negative.
3. Recall that the reference angle for 225° is 45°, since 225° - 180° = 45°.
4. Use the fact that \( \sin 45^\circ = \frac{\sqrt{2}}{2} \). Since 225° is in the third quadrant, \( \sin 225^\circ = -\frac{\sqrt{2}}{2} \).
5. Calculate \( \csc 225^\circ = \frac{1}{\sin 225^\circ} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2} \).