Question

What is the trigonometric form of `(7-i)`?

Answer 1

`5sqrt2(cos(0.142)-isin(0.142))`

Explanation:

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

That is `(x+yi)to[r(costheta+isintheta)]" where"`

`color(orange)"Reminder " 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)|)))`

For (7 - i) , x = 7 and y = - 1

`rArrr=sqrt(7^2+(-1)^2)=sqrt50=5sqrt2`

Now (7 - i) is in the 4th quadrant, so we must ensure that `theta` is in the 4th quadrant.

`theta=tan^-1(-1/7)=-0.142" in 4th quadrant"`

`rArr(7-i)=5sqrt2(cos(-0.142)+isin(-0.142))`

`color(orange)"Reminder"`

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

`rArr(7-i)=5sqrt2(cos(0.142)-isin(0.142))`