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Algebra Level 5

If a = cos ⁡ ( x ) + i sin ⁡ ( x ) a = \cos(x) + i\sin(x) and the equation a z 2 + z + 1 = 0 az^2 + z + 1 = 0 ( where z ∈ C z \in \mathbb{C} ) has a purely imaginary root , find the positive value of tan ⁡ ( x ) \tan(x) up to 3 decimal places.


The answer is 1.272.

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Aditya Narayan Sharma by Aditya Narayan Sharma , 21, India

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2 solutions

Considering the equation, z 2 z^{2} + z a \frac{z}{a} + 1 a \frac{1}{a} = 0

or, z 2 z^{2} + z ( c o s x + i s i n x ) − 1 (cosx + isinx)^{-1} + ( c o s x + i s i n x ) − 1 (cosx+isinx)^{-1} = 0

or, z 2 z^{2} + z(cosx - isinx) + cosx - isinx = 0 [By De-Moivre's theorem]

Say z = iy [as it is purely imaginary] ,

or, - y 2 y^{2} + iy(cosx - isinx) + cosx - isinx = 0

or, y 2 y^{2} - iy(cosx - isinx) + isinx - cosx = 0

or, [ y 2 y^{2} - ysinx - cosx] + i[sinx - ycosx] = 0 + i.0 Therefore,

Comparing The Imaginary Part We get y = tanx and putting it in the real part,

t a n 2 x tan^{2}x - tanx.sinx - cosx = 0

Taking t a n 2 x tan^{2}x = b and expressing sinx and cosx in tan we get the equation,

b 3 b^{3} - 2b - 1 = 0

Solving we get tanx = b \sqrt{b} = 5 + 1 2 \sqrt\frac{{\sqrt{5}+1}}{2}

Therefore tanx = 1.272 (approx)

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Moderator note:

That's an interesting problem! How did you set it up?

That's an interesting problem! How did you set it up?

Calvin Lin Staff - 5 years, 3 months ago

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Thank you sir ! I just came by it while going for an investigation with complex number equations, :)

Aditya Narayan Sharma - 5 years, 3 months ago

amazing. :) I wish I can learn all these stuffs.

Arjay Onan - 5 years, 3 months ago

please note : Dear Mr. Aditya Sharma , your final answer is correct , but as you can see your final equation ;" as you wrote it " is :b³-2b-1=0 is not correct , the correct equation after substituting tan²x = b ...... is : b²-b-1=0 , Now solving this final equation will give √b = √(1+√5)/2)). so far ,the correct answer is 1.272 , as you solve , which is the correct answer to the above problem. "this is just , a remark.". the problem is beautiful as well as its solutoin . thank you v.much

Aziz Alasha - 4 years, 5 months ago
Aakash Khandelwal
Feb 21, 2016

Put x=ib. Where b belongs to set of real numbers.
now equate real and imaginary parts to zero.
cos(x)= the golden ratio.
tan(x)=2/(10-2*(5^0.5))^0.5
Which is approximately the answer.



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cos(x) cannot be the golden ratio since the golden ratio is larger than one

Tom Van Lier - 5 years, 3 months ago

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