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I understand the reason that

$$\cos(\tfrac{\pi}{3} + 2\pi N) = \cos (\tfrac{\pi}{3})$$

is because if a line formimg an angle of $\pi/3$ with the horizontal makes $N$ complete revolutions, it still forms an angle of $\pi/3$.

With this understanding, if a line forms an angle of $120$ degree with the horizontal, it also forms an angle of $-60$ degree with the horizontal (negative because the direction from the negative $x$-axis to the line would be clockwise); hence, $$\sin (120°) = \sin (-60°)$$

But we know these two values are not equal.

Could you please explain where the error is coming from?

Thank you

Gary
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    One is positive and the other is negative: $\sin(2 \pi/3) = \sin(\pi/3) = -\sin(-\pi/3)$. – Martin R Nov 03 '22 at 08:53
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    Also: https://math.stackexchange.com/q/741255/42969, https://math.stackexchange.com/q/4399197/42969 – Martin R Nov 03 '22 at 08:57
  • @MartinR I don't think any of those really address explicitly the actual misconception at the heart of this question. It's buried down in some of the answers, but I don't think they will find it there if they haven't already figured it out from reading their own textbook. – Arthur Nov 03 '22 at 08:58
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    @Arthur: I disagree. “... by definition* $\cos \theta$ and $\sin \theta$ are precisely the coordinates of the point $P(x,y)$ on the unitary circle such that ray OP forms an angle $\theta$ with positive $x$ axis ...”* is not buried, but right at the start of the accepted answer to the suggested duplicate. – Martin R Nov 03 '22 at 09:02
  • @MartinR "Buried" might be a bit loose, fine. I still think it's a bit much to expect the OP to sift through the veritable information dump that those links represent and be able to pinpoint exactly the sentence they need when clearly they have already missed it before. – Arthur Nov 03 '22 at 09:05
  • "if a line forms an angle of 120 degree with the horizontal, it also forms an angle of −60 degree with the horizontal (negative because the direction from the negative x-axis to the line would be clockwise)" There's your answer. trig functions are not just determined by the line they are on but also by the direction of the line. The sine of $120$ is the point as you travel the line UPWARD and TO THE LEFT. It doesn't matter that if you travel the exact same line downward and to the right you get another value. – fleablood Nov 03 '22 at 09:13

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Angles in the unit circle when talking about trigonometric functions aren't measured against the line that is the $x$-axis. They are measured against the ray that is positive half of the $x$-axis. So your $\frac23\pi$ angle is also a $-\frac43\pi$ angle, but it is not a $-\frac13\pi$ angle ($-\frac13\pi$ is an angle that points down and to the right).

Arthur
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