Question

Which of the following expressions is not equivalent to the quotient of sqrt2(cos45° + isin45°) and sqrt2(cos315° + isin315°)?

Answer 1

Answer:

-i

Step-by-step explanation:

Given are two complex numbers as

$z1 = \sqrt{2} (cos 135+isin 135)z2 = \sqrt{2} (cos 315+isin315)$

To find quotient

We can use Demoivre theorem for products and quotients here

Quotient would be equal to

$\frac{z1}{z2} =\frac{\sqrt{2}(cos45+isin45) }{\sqrt{2}(cos315+isin315) } \\=cos (45-135)+isin(45-135)\\=cos90-isin90\\=-i$

Hence quotient = -i

Answer 2

Answer with explanation:

If, A complex number, Z=a + i b

then, Z can be written as,=|r| (Cos A + i Sin A), where |r|=Modulus of Complex number which is equal to

$|r|=\sqrt{a^2+b^2}\\\\A=\ Tan^{-1}(\frac{b}{a})\\\\a=r \ Cos A\\\\b=r \ Sin A\\\\Z=|r|e^{iA}$

The expression which is equivalent to √2[Cos 45° +i Sin 45°] is,

   $\Rightarrow \sqrt{2}( \ Cos45^{\circ} + i\ Sin 45^{\circ})\\\\\Rightarrow \sqrt{2}(\frac{1}{\sqrt{2}} +i\frac{1}{\sqrt{2}} )\\\\\Rightarrow \sqrt{2}\times\frac{1+i}{\sqrt{2}}\\\\=1+ i$

And , the expression which is equivalent to,√2[Cos 315° +i Sin 315°] is

 .  $\Rightarrow \sqrt{2}( \ Cos315^{\circ} + i\ Sin 315^{\circ})\\\\\Rightarrow \sqrt{2}( \ Cos45^{\circ} - i\ Sin 45^{\circ})\\\\\Rightarrow \sqrt{2}(\frac{1}{\sqrt{2}} -i\frac{1}{\sqrt{2}} )\\\\\Rightarrow \sqrt{2}\times\frac{1-i}{\sqrt{2}}\\\\=1- i$

→ Cos (360°-45°)=Cos 45°

→Sin (360° -45°)= -Sin 45°