If theta is an angle in standard position, state in what quadrants its terminal side can lie if theta is equal to -415 degrees?
get it positive: 720-415=305
looks like quadrant IV.
To determine in which quadrants the terminal side of an angle lies, you can follow these steps:
1. Start by placing the initial side of the angle on the positive x-axis (rightward direction).
2. If the angle is positive, rotate counter-clockwise.
3. If the angle is negative, rotate clockwise.
Now, let's apply this to the given angle θ = -415 degrees:
1. Start by placing the initial side of the angle on the positive x-axis.
2. Since θ is negative (-415 degrees), we need to rotate clockwise.
3. To simplify the angle, subtract 360 degrees multiple times until you get an angle between 0 and 360 degrees. In this case, -415 degrees - 360 degrees = -775 degrees.
4. Continue subtracting 360 degrees: -775 degrees - 360 degrees = -1135 degrees.
5. Once again: -1135 degrees - 360 degrees = -1495 degrees.
6. Repeat: -1495 degrees - 360 degrees = -1855 degrees.
7. One more time: -1855 degrees - 360 degrees = -2215 degrees.
8. Finally, -2215 degrees - 360 degrees = -2575 degrees.
After going through these steps, you now have the angle -2575 degrees, which is equivalent to -415 degrees.
Since -415 degrees is a full revolution beyond -360 degrees, it lies in the same quadrant as -415 degrees + 360 degrees = -55 degrees.
-55 degrees is in the fourth quadrant since it is between -90 degrees and -180 degrees.
Therefore, the terminal side of an angle of -415 degrees lies in the fourth quadrant.