Multiple Choice
Evaluate each expression.
9. Unit Circle
Reciprocal Trigonometric Functions on the Unit Circle
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Evaluate each expression.
A
B
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D
Verified step by step guidance1
First, understand that the cosecant function, \( \csc \theta \), is the reciprocal of the sine function, \( \sin \theta \). Therefore, \( \csc 225^\circ = \frac{1}{\sin 225^\circ} \).
Next, determine the reference angle for \( 225^\circ \). Since \( 225^\circ \) is in the third quadrant, the reference angle is \( 225^\circ - 180^\circ = 45^\circ \).
Recall that in the third quadrant, the sine function is negative. Therefore, \( \sin 225^\circ = -\sin 45^\circ \).
The value of \( \sin 45^\circ \) is \( \frac{\sqrt{2}}{2} \). Thus, \( \sin 225^\circ = -\frac{\sqrt{2}}{2} \).
Finally, calculate \( \csc 225^\circ = \frac{1}{\sin 225^\circ} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\sqrt{2} \).
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