Problem 44 - Precalculus: A Concise Course

Question

Determine the quadrant in which each angle lies. (a) \(-260^{\circ}\) (b) \(-3.4^{\circ}\)

Step-by-Step Solution

Answer

The angle \(-260^{\circ}\) lies in Quadrant I, and the angle \(-3.4^{\circ}\) lies in Quadrant IV.

Step 1: Determine Quadrant 1

For the angle of \(-260^{\circ}\), positive angles are usually measured counterclockwise from the positive x-axis while negative angles are measured clockwise. The magnitude of \(-260^{\circ}\) exceeds \(180^{\circ}\) (since \(260 > 180\)), indicating that the angle has passed Quadrant II, but has not completed a full rotation of \(360^{\circ}\). Thus, \(-260^{\circ}\) falls in Quadrant I.

Step 2: Determine Quadrant 2

Now consider the angle of \(-3.4^{\circ}\). Again, negative angles are measured clockwise from the positive x-axis and since no full rotation of \(360^{\circ}\) has been completed, we can observe that the angle lies in the short interval between the positive x-axis and mechanism starting to move to the negative x-axis. Thus, the angle \(-3.4^{\circ}\) falls in Quadrant IV.