Problem in finding a general solution

  • #1
navneet9431
Gold Member
107
9

Homework Statement


thumbnail_IMG_20180713_225234 (1).jpg


Homework Equations


General Formula for Tan(a)=Tan(b)
gif.gif


The Attempt at a Solution


See the question I have uploaded.

I have tried solving it this way,

Firstly I applied the Quadratic Formula to get,

gif.gif


Now we have two cases,

CASE-1

When
12%29%3D2-%5Csqrt3.gif


So General Formula here will be,

12.gif


Now, CASE-2

when
12%29%3D-%282+%5Csqrt3%29.gif


So General Formula here will be
12.gif


I do not know what should I do next to get the answer? Please tell me how to proceed Further.

The answer given in the key is the option (C).

I will be thankful for any help!
 

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Answers and Replies

  • #2
BvU
Science Advisor
Homework Helper
15,030
4,081
Hello,

So you need to combine ##\pi (n+ {1\over 12}) ## and ##\pi (n - {5\over 12}) ## . Leave the ##\pi## outside the brackets and try a few n. The pattern emerges !
 
  • #3
Delta2
Homework Helper
Insights Author
Gold Member
5,020
2,088
if the 6n looks scary in the option C, it is actually just ##n/2## cause ##a=(6n+1)\frac{\pi}{12}=\frac{n}{2}\pi+\frac{\pi}{12}##. From this very last expression for a, what do you get if you put
1) n=even=2k
2) n=odd=2k+1
 
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Likes ehild and scottdave
  • #5
ehild
Homework Helper
15,543
1,915
I think the problem is solved by @Delta2's hint, so I may come with a very simple solution of the equation tan2(α)+2√3 tan(α)=1,which can be rearranged to 1-tan2(α)=2√3 tan(α).
nπ/2 in the offered solutions suggests to solve the equation for 2α. The double-angle formula is ##\tan(2α)=\frac{2\tan(α)}{1-\tan^2(α)}##, that is ##\tan(2α)=\frac{2\tan(α)}{2\sqrt3 \tan(α)}=\frac{1}{\sqrt 3}##, that is, 2α=π/6+kπ and α=π/12+kπ/2.
 

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