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Chapter 5: Problem 47

Evaluate the trigonometric function of the quadrant angle, if possible. $$\csc \frac{3 \pi}{2}$$

Short Answer

Expert verified
\(\csc \frac{3 \pi}{2}\) is undefined.

Step by step solution

01

Recall the definition of the cosecant function.

The cosecant function is defined as the reciprocal of the sine function, i.e. \(\csc x = \frac{1}{\sin x}\).
02

Determine the value of the sine function at the given angle.

The angle given is \(\frac{3 \pi}{2}\) which is associated with the point (-1,0) on the unit circle. The sine of this angle can be determined based on the y-coordinate of the point on the unit circle, which in this case is 0. So, \(\sin \frac{3 \pi}{2} = 0\)
03

Subtitute the value of the sine function in the cosecant function.

Substitute the value of \(\sin \frac{3 \pi}{2}\) into the definition of the cosecant function. This yields: \(\csc \frac{3 \pi}{2} = \frac{1}{\sin \frac{3 \pi}{2}} = \frac{1}{0}\)
04

Evaluate the expression

As division by zero is undefined in the real number system, the expression \(\csc \frac{3 \pi}{2}\) is also undefined.

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