"to convert from"color(blue)" complex to trig. form"to convert from complex to trig. form
"that is "x+yitor(costheta+isintheta)" using"that is x+yi→r(cosθ+isinθ) using
•color(white)(x)r=sqrt(x^2+y^2)∙xr=√x2+y2
•color(white)(x)theta=tan^-1(y/x)color(white)(x);-pi< theta <=pi∙xθ=tan−1(yx)x;−π<θ≤π
"here "x=6" and "y=-7here x=6 and y=−7
rArrr=sqrt(6^2+(-7)^2)=sqrt85⇒r=√62+(−7)2=√85
6-7i" is in the fourth quadrant so we must ensure that "theta6−7i is in the fourth quadrant so we must ensure that θ
"is in the fourth quadrant"is in the fourth quadrant
rArrtheta=tan^-1(7/6)=0.862larrcolor(red)" related acute angle"⇒θ=tan−1(76)=0.862← related acute angle
rArrtheta=-0.862larrcolor(red)" in fourth quadrant"⇒θ=−0.862← in fourth quadrant
rArr6-7i=sqrt85(cos(-0.862)+isin(-0.862))⇒6−7i=√85(cos(−0.862)+isin(−0.862))
[cos(-0.862=cos(0.862);sin(-0.862)=-sin(0.862)][cos(−0.862=cos(0.862);sin(−0.862)=−sin(0.862)]
rArr6-7i=sqrt85(cos(0.862)-isin(0.862))⇒6−7i=√85(cos(0.862)−isin(0.862))
