The A + iB form of frac{(cos x + isin x)(cos | vedclass

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

Question

(A) Given expression: $Z = \frac{(\cos x + i\sin x)(\cos y + i\sin y)}{(\cot u + i)(1 + i	an v)}$Using Euler's formula,the numerator is $e^{i(x+y)}$.

Vedclass · Accepted Answer

$\sin u \cos v [\cos (x + y - u - v) + i\sin (x + y - u - v)]$

Vedclass · Answer

(A) Given expression: $Z = \frac{(\cos x + i\sin x)(\cos y + i\sin y)}{(\cot u + i)(1 + i	an v)}$Using Euler's formula,the numerator is $e^{i(x+y)}$.For the denominator:$(\cot u + i) = \frac{\cos u + i\sin u}{\sin u} = \frac{e^{iu}}{\sin u}$$(1 + i	an v) = \frac{\cos v + i\sin v}{\cos v} = \frac{e^{iv}}{\cos v}$Substituting these into the expression:$Z = \frac{e^{i(x+y)}}{\frac{e^{iu}}{\sin u} \cdot \frac{e^{iv}}{\cos v}} = \sin u \cos v \cdot \frac{e^{i(x+y)}}{e^{i(u+v)}}$$Z = \sin u \cos v \cdot e^{i(x+y-u-v)}$$Z = \sin u \cos v [\cos(x+y-u-v) + i\sin(x+y-u-v)]$

List-$I$	List-$II$
$(i)$ $a \bar{a}$	$(A)$ $-\frac{\pi}{3}$
$(ii)$ $\arg \left(\frac{1}{\bar{a}}\right)$	$(B)$ $-i \sqrt{3}$
$(iii)$ $a - \bar{a}$	$(C)$ $2i / \sqrt{3}$
$(iv)$ $\operatorname{Im}\left(\frac{4}{3a}\right)$	$(D)$ $1$
	$(E)$ $\pi / 3$
	$(F)$ $\frac{2}{\sqrt{3}}$

The $A + iB$ form of $\frac{(\cos x + i\sin x)(\cos y + i\sin y)}{(\cot u + i)(1 + i\tan v)}$ is

Similar Questions

If $z = \frac{\sqrt{3}}{2} + \frac{i}{2}$ where $i = \sqrt{-1}$,then $(1 + iz + z^5 + iz^8)^9$ is equal to:

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^2+|z|=0$,where $z \in \mathbb{C}$. Then $4(\alpha^2+\beta^2)$ is equal to:

Let $A = \left\{ \frac{1967 + 1686 i \sin \theta}{7 - 3 i \cos \theta} : \theta \in R \right\}$. If $A$ contains exactly one positive integer $n$,then the value of $n$ is

Let $\alpha, \beta$ be the roots of the equation $x^2-x+2=0$ with $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Then $\alpha^6+\alpha^4+\beta^4-5 \alpha^2$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module