For the rotation minus, start fraction, 80, pi, divided by, 11, end fraction− 1180π​ , find the coterminal angle from 0, is less than or equal to, theta, is less than, 2, pi0≤θ<2π, the quadrant, and the reference angle.

To find the coterminal angle, we need to add or subtract multiples of 2π until we get an angle that is between 0 and 2π.

Given angle is -1180π/11.

Step 1: Convert the angle to positive by adding 2π until it falls within the desired range.

Since 2π is approximately 6.28, we can multiply this by a number that will get us close to 1180/11, which is approximately 107.27.

If we multiply 6.28 by 17, we get approximately 106.76.

So, let's add 17*2π to the given angle:

-1180π/11 + 17*2π = -1180π/11 + 34π = (-1180π + 374π)/11 = -806π/11

This is still a negative angle, so we need to add 2π again.

-806π/11 + 2π = -806π/11 + 22π/11 = -784π/11

This is still negative, so we continue this process until we get a positive angle.

After repeating this process 36 times, we get the angle 8π/11 which is in the range 0≤θ<2π.

Step 2: Determine the quadrant of the angle.

Since 8π/11 is less than π/2, it is in the first quadrant.

Step 3: Find the reference angle.

The reference angle is the acute angle that the terminal side of the given angle makes with the x-axis. Since the angle is in the first quadrant, the reference angle is the angle itself, which is 8π/11.