If {tan ^{ - 1}}(alpha + ibeta ) = x + iy, th | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given ${	an ^{ - 1}}(\alpha + i\beta ) = x + iy$. Taking the tangent of both sides,we get $\alpha + i\beta = 	an (x + iy) \dots (i)$. Taking the

Vedclass · Accepted Answer

$\frac{1}{2}{	an ^{ - 1}}\left( {\frac{{2\alpha }}{{1 - {\alpha ^2} - {\beta ^2}}}} ight)$

Vedclass · Answer

(A) Given ${	an ^{ - 1}}(\alpha + i\beta ) = x + iy$. Taking the tangent of both sides,we get $\alpha + i\beta = 	an (x + iy) \dots (i)$. Taking the complex conjugate of both sides,we get $\alpha - i\beta = 	an (x - iy) \dots (ii)$. Using the formula $	an (A + B) = \frac{	an A + 	an B}{1 - 	an A 	an B}$,we have: $	an 2x = 	an [(x + iy) + (x - iy)] = \frac{	an (x + iy) + 	an (x - iy)}{1 - 	an (x + iy) 	an (x - iy)}$. Substituting the values from $(i)$ and $(ii)$: $	an 2x = \frac{(\alpha + i\beta ) + (\alpha - i\beta )}{1 - (\alpha + i\beta )(\alpha - i\beta )} = \frac{2\alpha }{1 - (\alpha ^2 + \beta ^2)}$. Therefore,$2x = 	an ^{-1} \left( \frac{2\alpha }{1 - \alpha ^2 - \beta ^2} ight)$. Thus,$x = \frac{1}{2} 	an ^{-1} \left( \frac{2\alpha }{1 - \alpha ^2 - \beta ^2} ight)$.

If ${\tan ^{ - 1}}(\alpha + i\beta ) = x + iy,$ then $x =$

Similar Questions

If $|z_1| = |z_2|$ and $\arg\left( \frac{z_1}{z_2} \right) = \pi$,then $z_1 + z_2$ is equal to

Let $z_1$ and $z_2$ be two complex numbers and roots of the equation $z^2 + az + b = 0$. If $O$ is the origin such that $OA = OB$ and $a^2 = \lambda b \cos^2 \frac{\alpha}{2}$,where $\alpha$ is the angle $\angle AOB$,then $\lambda$ is equal to:

Let $S = \{z = x + iy : \frac{2z - 3i}{4z + 2i} \text{ is a real number} \}$. Then which of the following is $NOT$ correct?

The equation $\overline{b}z + b\overline{z} = c$,where $b$ is a non-zero complex constant and $c$ is real,represents:

The point $z$ in the Argand plane moves such that $\operatorname{Re} \left( \frac{iz + 1}{iz - 1} \right) = 2$. Then the locus of $z$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module