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There are certain identities that help us to determine the values of trigonometric functions at $\dfrac{\pi}{2}+x \text{, } \pi-x$ etc. given the values of $\sin x, \cos x$.

Now, when we prove such identities, we usually take the value of $x$ to be in the interval $\Big (0, \dfrac{\pi}{2} \Big )$. Isn't it necessary to prove the identities by taking the value of $x$ in all $4$ quadrants individually and then arriving at the outcome? If not, then why not?

Pardon me if this sounds silly.

Thanks!

Rajdeep Sindhu
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1 Answers1

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If the basis of your proofs are from Euler's identity $e^{i \theta} = \cos \theta + i \sin \theta$ then the quadrant becomes irrelevant.

Also consider the identities $\cos \theta = \dfrac {e^{i \theta} + e^{-i \theta} } 2$ and $\sin \theta = \dfrac {e^{i \theta} - e^{-i \theta} } 2$.

Prime Mover
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  • The concept of complex numbers has not been introduced to me yet. In my Mathematics textbook, they have been proved using a unit circle and finding the coordinates of the points where the allied angles intersect with the unit circle. Shall I elaborate? – Rajdeep Sindhu May 27 '20 at 15:00
  • No, I understand the form. My answer would be: yes, I would apply a geometrical argument to all 4 quadrants. A proof that did not would probably not satisfy me if I were doing it that way. BTW not a silly question at all. – Prime Mover May 27 '20 at 15:02
  • It doesn't satisfy me too! Thanks – Rajdeep Sindhu May 27 '20 at 15:04
  • By the way, the identities that you mention in your answer, they are related to complex numbers, right? – Rajdeep Sindhu May 27 '20 at 15:04
  • Yes they are -- they crop up when you try to define trig functions for complex numbers. By that stage, their origin as ratios of sides of triangles has got lost in the distance. – Prime Mover May 27 '20 at 15:07
  • Thanks! That was really helpful. By the way, are you familiar with the algorithm for finding the values of trigonometric functions that basically just summarizes the identities I refer to? Do you think that that algorithm really is necessary? I discussed it in a question here : https://math.stackexchange.com/questions/3676830/why-do-we-use-this-complicated-algorithm-for-finding-values-of-trigonometric-fun – Rajdeep Sindhu May 27 '20 at 15:10
  • Never used it. I mistrust algorithms and mnemonics because then you are tempted not to learn the reasons and the true nature of a concept at the expense of a list of rules. Bad way to study mathematics. I learned at the start what a cosine and sine are, and whenever I need to remember one of the rules I flash up a little geometric diagram in my head. Cosine goes across and sine goes up and down. (Remember that "sine" was a mistranslation via Arabic of the Sanskrit word meaning "bowstring" and you are already there.) – Prime Mover May 27 '20 at 15:30
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    You have no idea about how much troubled and confused I was due to that algorithm. While I find some algorithms helpful, I hate all mnemonics like the one for remembering the signs for trigonometric functions in various quadrant All School To College. I wish there was a 'follow' option on this website. Thanks, again! PS : The 'algorithm' is useful for making a computer program though – Rajdeep Sindhu May 27 '20 at 15:36