Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

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

Step 1: Recall the definition of the cosecant function.

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

Step 2: 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, \(\sin \frac{3 \pi}{2} = 0\)

Step 3: 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: \(\csc \frac{3 \pi}{2} = \frac{1}{\sin \frac{3 \pi}{2}} = \frac{1}{0}\)

Step 4: Evaluate the expression

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