ISI2019-MMA-9

Question

$(\cos 100^\circ +  i \sin 100^\circ)(\cos 0^\circ + i \sin 110^\circ)$ is equal to

• A. $\frac{1}{2}(\sqrt3 – i)$
• B. $\frac{1}{2}(-\sqrt3 – i)$
• C. $\frac{1}{2}(-\sqrt3 +i)$
• D. $\frac{1}{2}(\sqrt3 + i)$

Comments

yes B)

Answer 1

$cos\Theta + isin\Theta =e^{i\Theta }$

Therefore, $(\cos100+i\sin100) \times (\cos110+i\sin110)==e^{i100} \times e^{i110} =e^{i210}$ =  $(\cos210+i\sin210)=cos(180+30) + isin(180+30)=cos180cos30 -sin180sin30 + i(sin180cos30+cos180sin30) = -\frac{\sqrt3}{2} +0+i(0-\frac{1}{2})= -\frac{\sqrt3}{2} - \frac{i}{2} = \frac{1}{2}(-\sqrt3-i)$