Problem 62 - Precalculus Essentials

Question

Find a positive angle less than \(360^{\circ}\) or \(2 \pi\) that is coterminal with the given angle. $$-760^{\circ}$$

Step-by-Step Solution

Answer

The positive angle that is coterminal with -760 degrees is 320 degrees.

Step 1: Understanding Co-terminal Angles

The angles are coterminal if they share the same initial side and terminal sides. In this case, by adding or subtracting multiples of 360 degrees, the co-terminal angle can be found. Because 360 degrees represent a complete circle, adding or subtracting 360 degrees from a given angle results in an angle that is coterminal with the original angle.

Step 2: Consider the Negative Angle

The given angle is -760 degrees, which means it is rotated in the clockwise direction. But we need the positive value that shows an anti-clockwise rotation. So we need to add 360 degrees repeatedly until the angle is in the range of 0 and 360 degrees.

Step 3: Adding 360 Degrees

Starting with -760 degrees, add 360 degrees: -760 degrees + 360 degrees = -400 degrees. Since -400 degrees is still less than 0, continue to add 360 degrees: -400 degrees + 360 degrees = -40 degrees. Once again, -40 degrees is less than 0, so add more 360 degrees: -40 degrees + 360 degrees = 320 degrees.

Step 4: Verifying the Result

Now the angle 320 degrees is a positive angle and is also within the 0 to 360 degrees range. Therefore, 320 degrees is the correct answer.