Why is -20pi/3 conteminal with 4pi/3
Perhaps you can see it better if we change it to degrees
-20π/3 radians = -1200°
This would be 3 complete clockwise rotations, ending up at -120°
We can also reach -120° by going counter-clockwise for 240°
Well, and how about that?
Isn't 4pi/3 radians = 240°
To determine if two angles are coterminal, we need to check if they end at the same position on the unit circle. In this case, the angles in question are -20π/3 and 4π/3.
To compare these angles, we need to convert them to the same range by adding or subtracting a multiple of 2π. Let's convert -20π/3 to radians within the range of [0, 2π]:
-20π/3 = -6π + (2π/3)
Since 2π is equivalent to a full revolution, we can subtract 6π (or add 6π) repeatedly until we get an angle within the desired range:
-6π + (2π/3) = -4π + (2π/3) = -2π + (2π/3) = 0 + (2π/3) = 2π/3
Now we have the angle 2π/3. Comparing this with 4π/3, we can see that they both end at the same position on the unit circle.
Therefore, -20π/3 is coterminal with 4π/3.
To determine if -20π/3 is coterminal with 4π/3, we need to check if they represent the same angle.
Two angles are coterminal if they terminate at the same position on the unit circle.
To find coterminal angles, we can use the fact that adding or subtracting a full revolution (2π radians or 360 degrees) does not change the reference angle.
In this case, the reference angle for both -20π/3 and 4π/3 is π/3.
Let's calculate the coterminal angle for -20π/3:
-20π/3 + 2π = -20π/3 + 6π/3 = -14π/3
Now, let's compare -14π/3 and 4π/3:
-14π/3 is not equal to 4π/3. Therefore, -20π/3 is not coterminal with 4π/3.
In summary, -20π/3 is not coterminal with 4π/3 because they do not represent the same angle when measured in the same direction on the unit circle.