Problem 42 - Precalculus: A Concise Course

Question

Determine the quadrant in which each angle lies. (a) \(8.3^{\circ}\) (b) \(257^{\circ} 30^{\prime}\)

Step-by-Step Solution

Answer

The angle \(8.3^{\circ}\) lies in Quadrant I and the angle \(257.5^{\circ}\) lies in Quadrant III.

Step 1: Determine the quadrant for \(8.3^{\circ}\)

This part is quite straightforward since the angle 8.3 falls within Quadrant I, given that it is between \(0^{\circ}\) and \(90^{\circ}\).

Step 2: Convert \(257^{\circ} 30^{\prime}\) into degrees

In mathematics, a prime is used to denote minutes, which is a way of measuring angles. We've been given \(257^{\circ} 30^{\prime}\) - to convert this into degrees, we remember that 1 degree is equal to 60 minutes. So, \(30^{\prime}\) is equal to \(30/60 = 0.5^{\circ}\). Therefore, our angle in degrees is \(257^{\circ} + 0.5^{\circ} = 257.5^{\circ}\).

Step 3: Determine the quadrant for \(257.5^{\circ}\)

Now that we have the angle in degrees, we can determine the quadrant. The angle \(257.5^{\circ}\) is greater than \(180^{\circ}\) but less than \(270^{\circ}\), therefore it falls within Quadrant III.