For the rotation start fraction, 28, pi, divided by, 3, end fraction 328π , 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.
Question
For the rotation start fraction, 28, pi, divided by, 3, end fraction 328π , 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.
Solution
The given rotation is 328π/3. To find the coterminal angle between 0 and 2π, we need to reduce the given rotation until it falls within this range.
Step 1: Find how many full rotations (2π) are in 328π/3. Divide 328π/3 by 2π to get 328/6 = 54.67. This means there are 54 full rotations with a bit left over.
Step 2: Subtract the full rotations from the original rotation. Subtract 54*2π from 328π/3 to get the coterminal angle. This gives us 328π/3 - 108π = 4π/3.
So, the coterminal angle 0 ≤ θ < 2π is 4π/3.
Step 3: Determine the quadrant. The angle 4π/3 is in the third quadrant because it is greater than π and less than 3π/2.
Step 4: Find the reference angle. The reference angle is the acute angle the terminal side of the angle makes with the x-axis. For an angle in the third quadrant, subtract π from the angle to get the reference angle. So, the reference angle is 4π/3 - π = π/3.
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