Question

Need Some Urgent Help!!

Rewrite the complex number in rectangular form, a + bi.
z=(sqrt 2)cis(105 deg)=?

A. ((1+(sqrt 3))/2)+((1+(sqrt 3))/2)i
B. ((1+(sqrt 3))/2)+((1-(sqrt 3))/2)i
C. ((1-(sqrt 3))/2)+((1+(sqrt 3))/2)i
D. ((1-(sqrt 3))/2)+((1-(sqrt 3))/2)i

Work so far: √2∠105 = √2 cos105 + √2sin105i = -0.36 + 1.63i

Dont really know where to go from here, it does't match any of the answers..

Answer 1

\[\large\rm \sqrt2~cis(105)\quad=\quad \sqrt2~cis(45+60)\]Hmm I guess maybe we have to apply our Angle Sum Formulas for Sine and Cosine.

Answer 2

\[\large\rm =\sqrt2\left[\color{royalblue}{\cos(45+60)}+i \sin(45+60)\right]\]Apply our Cosine Angle Addition Formula,\[\large\rm =\sqrt2\left[\color{royalblue}{\cos45\cos60-\sin45\sin60}+i \sin(45+60)\right]\]

Answer 3

And you'll want to apply the Sine Angle Addition Formula to the other term, and then distribute the i,
and then turn all those into special values using your unit circle.
and then combine like-terms.

It's going to be a bit of a doozy...

Make sense?

Answer 4

i think so thanks for helping!