Question

How do you write `(2)-(2isqrt3)` in trigonometric form?

Answer 1

`4(cos(pi/3)-isin(pi/3))`

Explanation:

To convert from `color(blue)"complex to trigonometric form"`

That is `x+yitor(costheta+isintheta)`

`color(red)(|bar(ul(color(white)(a/a)color(black)(r=sqrt(x^2+y^2))color(white)(a/a)|)))" and " color(red)(|bar(ul(color(white)(a/a)color(black)(theta=tan^-1(y/x))color(white)(a/a)|)))`

here x = 2 and `y=-2sqrt3`

`rArrr=sqrt(2^2+(2sqrt3)^2)=sqrt(4+12)=sqrt16=4`

Now `2-2sqrt3 i` is in the 4th quadrant so we must ensure that `theta` is in the 4th quadrant.

`theta=tan^-1((-2sqrt3)/2)=tan^-1(-sqrt3)`

`=-pi/3" in 4th quadrant"`

`rArr2-2sqrt3 i=4(cos(-pi/3)+isin(-pi/3))`

which can also be expressed as `4(cos(pi/3)-isin(pi/3))`

Since `color(red)(|bar(ul(color(white)(a/a)color(black)(cos(-theta)=costheta" and " sin(-theta)=-sintheta)color(white)(a/a)|)))`