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How come point B make (cos $\alpha$ , sin $\alpha$) for a unit circle

If it was in the first quadrant i could figure it out.But since it is in the second quadrant how come it happen.You cant directly go for a right angle triangle.

Angle $\angle BAD$ is making 180-$\alpha$ degree ..But i cant figure it out the more useful facts on how its making cos $\alpha$, and sin $\alpha$ as coordinates of point B.

I am trying to understand the proof in page 16 ..but became clueless as it is taken for granted

https://ncert.nic.in/textbook/pdf/kemh103.pdf

Please help me...If this is a stupid question ..please have some kindness towards me. n

Midhun Raj
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  • For angles larger than $90°$, $\cos$ and $\sin$ are defined to be the coordinates in question, for the sole reason that it works for angles smaller than $90°$ and ends up being useful. – Vercassivelaunos Aug 08 '22 at 04:09
  • @Vercassivelaunos ..you might be correct but i am trying to understand the proof of cos(x+y)=coscosy-sinx siny ..There it is taken for granted ...16 th page of this pdf ..it is stated https://ncert.nic.in/textbook/pdf/kemh103.pdf ...but not given in detail how it came about – Midhun Raj Aug 08 '22 at 04:13
  • @Vercassivelaunos ..Thanks for trying to help me. ..but i think you could make it more clear ...sorry form my side ...I didnt fully understand the concept as till now – Midhun Raj Aug 08 '22 at 04:14
  • Just for clarification: what does $\cos(120°)$ mean to you? – Vercassivelaunos Aug 08 '22 at 04:18
  • cos(180-60) = -cos(60) ... – Midhun Raj Aug 08 '22 at 04:20
  • @Vercassivelaunos ..but what does that have relevance here...Can you please explain it more...Sorry for my ignorence – Midhun Raj Aug 08 '22 at 04:21
  • @MidhunRaj: Perhaps this old answer of mine will help. – Blue Aug 08 '22 at 04:28
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    @Blue ..Thanks a ton dear ..That is an excellent awesome write up.Just had a glance through it..and felt a goose bumps ..and high doze of happiness on that beautiful write up..Now i will spend sometime in trying to understand it fully..Thanks a ton – Midhun Raj Aug 08 '22 at 04:35
  • That's not what $\cos(120°)$ means. It's just a way to calculate it. The symbols we use have meaning, and the formulae we use (like $\cos(\theta+180°)=-\cos(\theta)$) are a consequence of that meaning. The meaning of the cosine is exactly the $x$-coordinate of the corresponding point on the unit circle. That's a first principle. It can't be derived. It's everything else that's derived from that first principle – Vercassivelaunos Aug 08 '22 at 10:23
  • @Vercassivelaunos: "The meaning of the cosine is exactly the $x$-coord[...] on the unit circle." .. What counts as a "first principle" is subjective, as there are competing defns of cosine. "$x$-coord on the unit circle" is one; a power series or soln to a differential eqn are others. I prefer to define cosine for acute angles via the right-triangle ratio, then (as linked above) to use relations from first-quadrant trig to push the boundaries of knowledge into the other quadrants. It's "inefficient" compared to starting w/the unit circle, but it serves the narrative of math as a journey. – Blue Aug 08 '22 at 12:18
  • @Blue True, there can be different first principles, but the fact that the textbook in question uses the unit circle to prove trig identities makes me assume that the author chose the unit circle as a first principle. Besides, most first principles can be used to narrate math as a journey, as long as you can motivate them. – Vercassivelaunos Aug 08 '22 at 16:21
  • @Blue and Vercassivelaunos: ..You guys are really amazing..Now i understood that that cosine in unit circle is a first principle(hearing this term for the first time) assumption and it to be taken for granted and are never to be questioned ...Thanks a lot. – Midhun Raj Aug 08 '22 at 16:37

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