complex number

Question

If one vertex of an equilateral triangle is 1+i and centroid is it's origin then other two vertices of triangle are?

Answer 1

DeMoivres theorem again. plot the point (1,1), the representation of 1+i. Connect to the origin ane draw a reference triangle. The hypotenuse is sqrt2 and the angle is 45. Writing this in polar form is 1+i = sqrt 2(cos 45 + i sin 45). since we're dealing with an equilateral triangle again, we need to add 360/3 or 120. so your other points are

scrt 2(cos 165 + i sin 165) and sqrt 2( cos 285 + i sin 285) and you're done. converting this back into rectangular form is a little more work.

you need the sum formulas if you really want the exact answers.

cos(a+b)= cos a*cos b - sin a * sin b

cos (45 +120) = cos 45*cos120 - sin 45*sin120 = (sqrt2 / 2)*(-1/2) - (sqrt2 /2)*(sqrt3 /2) = -(sqrt2 )/4 - (sqrt 6)/4.

 

You'd have to do the same for sin 165, cos 285, and sin 285 using the appropriate formulas and any 2 angles that add to 285, first one coming to my head is 240+45 but you could use 150 + 135.

 

I'm a little surprised that this was an equilateral triangle problem, a square works out so much nicer.