Converting a number to rectangular form

[Ry122]

Find the rectangular form of the complex number z=a+bi which has modulus 2 and principal argument theta = -pie/6
2cos(-pie/6) to get length of x
=sqrt3 What is the method used to get an answer in this form instead of decimal form (1.732) ?

 

[rock.freak667]

Use Euler's formula

 

[Ry122]

Could you please show me the steps?

 

[Dick]

The complex number with modulus r and argument theta is r*exp(i*theta)=r*(cos(theta)+i*sin(theta)). exp(i*theta)=cos(theta)+i*sin(theta) is Euler's formula.

 

[Ry122]

Isn't there a quicker way to do it?

 

[Dick]

?? If you followed that then the real part of z is x=r*cos(theta)=2*cos(-pi/6). What do you mean 'quicker'. I just wrote it down. I didn't compute anything. How could anything else be quicker?

 

[Ry122]

I have that in my initial post. I was just wanting to know how u get sqrt3 instead of 1.732.

 

[Dick]

Oh. Sorry. Draw an equilateral triangle with side 1. Then pi/6 is half of one of the angles (since they are all pi/3). So split the equilateral triangle into two right triangles with base 1/2 and hypotenuse 1. The remaining side is sqrt(3)/2. Use pythagoras. cos(pi/6) is that remaining side over the hypotenuse. So cos(pi/6)=sqrt(3)/2. So 2*cos(pi/6)=sqrt(3). Misunderstood your question.