If-tan-x-iy-a-bi-then-find-a-b-

Question Number 96290 by 175 last updated on 31/May/20

I f : \tan (x + i y) = a + b i

t h e n f i n d a, b

Commented by Tony Lin last updated on 31/May/20

t a n (x + i y)

= \frac{t a n x + t a n i y}{1 - t a n x t a n i y}

= \frac{t a n x + i t a n h y}{1 - t a n x \cdot i t a n h y}

= \frac{(t a n x - t a n h y i) (1 + t a n x t a n h y i)}{(1 - t a n x t a n h y i) (1 + t a n x t a n h y i)}

= \frac{t a n x + {t a n x t a n h}^{2} y}{1 + {t a n}^{2} {x t a n h}^{2} y} + \frac{{t a n}^{2} x t a n h y - t a n h y}{1 + {t a n}^{2} {x t a n h}^{2} y} i