There are four complex fourth roots to the number −16i. These can be expressed in polar form as z1=r1(cosθ1+isinθ1) , z2=r2(cosθ2+isinθ2), z3=r3(cosθ3+isinθ4) and z4=r4(cosθ4+isinθ4), where ri is a positive real number and 0∘≤θi<360∘. What is the value of θ1+θ2+θ3+θ4 (in degrees)?
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In polar form, -16i = 16 [cos(3π/2) + i sin(3π/2)]
So, De Moivre's Theorem yields
(-16i)^(1/4) = 16^(1/4) [cos((3π/2 + 2πk)/4) + i sin((3π/2 + 2πk)/4)]
................= 2 [cos(π(3 + 4k)/8) + i sin(π(3 + 4k)/8)], k = 0, 1, 2, 3.
Hence the sum of the angles equals
π(3 + 0)/8 + π(3 + 4)/8 + π(3 + 8)/8 + π(3 + 12)/8
= 36π/8
= 9π/2, or (9/2) * 360° = 1620°.
I hope this helps!
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Use the quadratic formulation for the equation written interior the type ax^2 + bx + c = 0. x = (-b + - sqrt (b^2 - 4ac) / 2a . . .the place a = coefficient of x^2 term, b = coefficient of x term , and c = the consistent. In our issue, a = 5, b = 2, and c = a million. x = (-2 + - sqrt (2^2 - 4(5)(a million)) / (2)(5) = (-2 + - sqrt (4 - 20)) / 10 = (-2 + - sqrt (-sixteen)) / 10 = (-2 + - sqrt ((sixteen)(-a million)) / 10 = (-2 + - 4i) / 10 = (-a million + - 2i) / 5