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Add to -8pi / 3 the smallest multiple of 2pi that results in a positive number:
-8pi / 3 + 4pi = 4pi / 3
You add or subtract 2π until the answer is between 0 and 2π.
Like for a, (first change 2π to 10π/5 so it has a common denominator)
do 19π/5 - 10π/5 = 9π/5 which is in your interval.
2π = 6π/3 so now add that to -16π/3. You get -10π/3, still too low so add 6π/3 again.
-4π/3 still too low, do it once more
2π / 3 ok now
2π = 22π/11 so the last one is fine as is.
and 69π/2? Well, 2π is 4π/2 and you'd have to subtract until tomorrow to get 69 down to what you want so divide by4π/2 and see what the remainder is. 69 ÷ 4 = 17 remainder 1 so you have 1π/2
or just π/2 for the answer.
Please let me know if you need any clarification. I'm always happy to answer your questions.
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