Determine-the-complex-number-z-such-that-z-2-tanx-icosx-in-the-form-z-p-ik-p-k-R-

Question Number 558 by 112358 last updated on 26/Jan/15

D e t e r m i n e t h e c o m p l e x n u m b e r

z s u c h t h a t z^{2} = t a n x + i c o s x,

i n t h e f o r m z = p + i k, p, k \in R .

Answered by prakash jain last updated on 26/Jan/15

{(p + i k)}^{2} = \tan x + i \cos x

(p^{2} - k^{2}) + 2 i p k = \tan x + i \cos x

\tan x = p^{2} - k^{2} \Rightarrow k^{2} = p^{2} - \tan x

\cos x = 2 p k

\cos^{2} x = 4 p^{2} (p^{2} - \tan x)

4 p^{4} - 4 p^{2} \tan x - \cos^{2} x = 0

p^{2} = \frac{4 \tan x \pm \sqrt{{16 \tan}^{2} x + {16 \cos}^{2} x}}{8}

p^{2} = \frac{\tan x + \sqrt{\tan^{2} x + \cos^{2} x}}{2}

k^{2} = p^{2} - \tan x

k^{2} = \frac{\sqrt{\tan^{2} x + \cos^{2} x} - \tan x}{2}

p = \pm \sqrt{\frac{\sqrt{\tan^{2} x + \cos^{2} +} \tan x}{2}}

q = \pm \sqrt{\frac{\sqrt{\tan^{2} x + \cos^{2} x} - \tan x}{2}}

Check

p^{2} - q^{2} = \tan x

2 p q = 2 \times \pm \sqrt{\frac{\sqrt{\tan^{2} x + \cos^{2} x} + \tan x}{2}} \times \pm \sqrt{\frac{\sqrt{\tan^{2} x + \cos^{2} x} + \tan x}{2}}

= (\pm \times \pm) \cdot 2 \cdot \sqrt{\frac{\tan^{2} x + \cos^{2} x - \tan^{2} x}{4}} = \cos x