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I am resolving some trigonometric equation systems. A typical system (the following one is invented) could be:

$\sin(x) + \cos(y) = 1.5$

$2\sin(x) + 3\cos(y) = 1$

After getting the values, you just go to the unit circle and find the trigonometric solution.

However, I stumbled upon a different equation system (again invented system, I am not interested in the solution itself):

$\cot(x) + \cot(y) = 2$

$2\cot(x) + 3\cot(y) = 3.2$

Which I do not know how to proceed, since I cannot compare to the typical values found in the unit circle (sin/cos). What should I proceed after obtaining the values for $x [\cot(x)]$ and $y [\cot(y)]$$?

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2 Answers 2

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From what you've written:
$-2cot(x)-2cot(y)=-4$
$2cot(x) + 3cot(y) = 3.2$
Therefore, $cot(y)=-0.8$ and $cot(x)=2.8$. You can then write $x$ and $y$ as inverse of these values. Does this answer your question?

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  • $\begingroup$ Thank you for your answer. I know how to solve the equation system and there is no difficult at all with that. But AFAIK, when you solve an equation system involving trigonometric identities you should show where x is 2.8 (taking your answer as example) in the unit circle (i.stack.imgur.com/r8uHr.gif). For example, if you get (another example) that sin(x) = 1, then you know that solves to x=90º. The problem comes when having a cot instead of a sin/cos. $\endgroup$
    – Raj
    Jan 17, 2014 at 18:18
  • $\begingroup$ It's not possible to write the answer in a nice way like some rational multiple of $\pi$. You can write the answers as $x=arccot2.8+k\pi, k\in \mathbb{Z}$. $\endgroup$
    – Spock
    Jan 17, 2014 at 18:25
  • $\begingroup$ I am not sure if this is what the teacher is expecting, but sounds like the only possibility, so choosing your answer as accepted, thank you! $\endgroup$
    – Raj
    Jan 17, 2014 at 18:38
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Recall that we can calculate $\cot\theta$ using $\sin\theta$ and $ \cos\theta$ for any angle $\theta$:

$$\cot \theta = \dfrac{\cos \theta}{\sin \theta}$$

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  • $\begingroup$ First of all, thanks for your answer. Yeah, I know this identity. My question is about how to finish the exercise itself. If I had a sine or a cosine, I would just go to check the unit circle for the value of x/y and give the result (e.g. x= mπ6+πm, m∈Z). But how to proceed here? $\endgroup$
    – Raj
    Jan 17, 2014 at 18:11
  • $\begingroup$ Your question isn't very specific nor very clear. I'm not clear what you asking. $\endgroup$
    – amWhy
    Jan 17, 2014 at 18:13
  • $\begingroup$ Sorry, I have some difficulties with English. I tried to clarify it in the other question, hope that solves the misunderstanding. $\endgroup$
    – Raj
    Jan 17, 2014 at 18:24

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