Suppose that the point P is the point on the unit circle that is associated by the wrapping function with the number 31. In which quadrant is the point P located?
31 is between 9.5pi and 10pi, so this number is in Quadrant IV.
To determine the quadrant in which the point P is located, we need to use the angle associated with the point P on the unit circle.
1. The wrapping function maps real numbers onto the unit circle by wrapping them within the range of 0 to 2π (or 0 to 360 degrees).
2. Since the number 31 is not within this range, we need to find its equivalent within the range.
3. To find the equivalent angle, we can subtract or add multiples of 2π until the number is within the range.
- In this case, we subtract multiples of 2π: 31 - 2π = 31 - 6.28 ≈ 24.72
4. Now, we have the equivalent angle of 24.72 associated with the point P on the unit circle.
5. To determine the quadrant, we can examine the sign of the trigonometric functions (sine and cosine) for this angle.
- For angles between 0 and π/2 (0 to 90 degrees), both sine and cosine are positive. This corresponds to the first quadrant.
- For angles between π/2 and π (90 to 180 degrees), sine is positive but cosine is negative. This corresponds to the second quadrant.
- For angles between π and 3π/2 (180 to 270 degrees), both sine and cosine are negative. This corresponds to the third quadrant.
- For angles between 3π/2 and 2π (270 to 360 degrees), sine is negative but cosine is positive. This corresponds to the fourth quadrant.
6. In this case, the equivalent angle of 24.72 falls between 0 and π/2, so the point P is located in the first quadrant.