Intervals on unit circle

[sharp ocean]

Given the unit circle and the interval (-2pi, pi), am I correct in that the interval represents all angles alongsie the red arrow, in the order of its direction?

[sharp ocean]

Anyone?

[minor hinge]

When you go from -2π to 0, you cover the entire circle as just rotated by an angle of 2π. When you go from 0 to π, you further cover a semicircle. So overall, you cover the entire circle.

Note that going from -2π to π is essentially the same as going from 0 to 3π, as the starting points are the same and you cover an angle of 3π in the counter-clockwise direction.

[sharp ocean]

That is what I initially thought. I am trying to solve |sin(x/2)| = 1 witihn the interval (-2pi,pi). The textbook states that the total solutions amount of solutions within the interval is 1. How is this possible in this case? pi/2 and 3pi/2 are both solutions within (-2pi,0) so the interval (-2pi,pi) should have 3possiblesolutions.I can't seem to make sense out of the answer...

[minor hinge]

First of all, π/2 and 3π/2 do NOT lie in the interval (-2π,0), but -π/2 and -3π/2 do.

Now let's see

[sharp ocean]

my bad

[minor hinge]

Since we have the absolute value function, sin(x/2) can either be -1 or 1. But you should keep in mind that the input is x/2 and not x.

For example, if sin(x/2) = 1, x/2 = π/2 is a possible answer, but we need to find the value of x and not x/2

Hence, x = π is a solution, NOT π/2

Since we have multiple possible values of x/2 (where sin(x/2) could be 1 or -1), find them, multiply by 2 to get x, and check which of them lie in the given interval (-2π,π)

I hope this helps. If I failed to explain something clearly, please let me know.

I would also like to add that since the interval (-2π,π) is open at both -2π and π, these values are not to be included. Just something to keep in mind.

[minor hinge]

If it makes it any easier to comprehend, put x = π in the expression, which gives us sin(π/2), which is indeed equal to 1.

[minor hinge]

@sharp ocean was this any helpful?

[daring geode]

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