Solve the complex equation z^3 + 32 + 32i(sqrt(3)) = 0
First thing to observe here, is that this equation is somewhat different to what you would find normally. This is a complex equation. How can you tell? The third term, ie 32i(sqrt(3)) contains the symbol for the complex numbers. It is really important to spot this as the first thing once you see such a question in your exam. This means we will be dealing with angles and argand diagrams! Regardless of what this equation represents, we still solve it with the same approach as any equation! Let's isolate the unknown term z.aThis would give, z3 = -32 - 32sqrt(3)i . Remember that complex numbers have a graphical representation; that's what makes them special! Let's draw an argand diagram to see this. Draw your x and y axis; and mark the spot (-32, -32sqrt(3) ). This forms a triangle on the lower left quadrant right? What's the next step then? The question would require us to represent our answer in the exponential format, ie z = rexp(iθ). The diagram then shows us that z^3 is a number formed by a triangle with base 32 and height 32root3. Let's use the standard trigonometric results to obtain a value for the hypotenuse (thus radius) and the angle it makes with the (negative) x axis. For the angle: lets use tan(theta) = opp/adj; leading to tan(theta) = 32root(3)/32 = root(3) . This is pi /3 by using arctan(root3).For the radius: lets use pythagoras' theorem to obtain that the hypotenuse is Sqrt( 32^2 + [32(root3)]^2) = 64. Another quicker way is to see that this is a standard 1-root3-2 triangle, which can be quicker to obtain its hypotenuse. Regardless, we now have two new pieces of information. The angle formed with the negative x-axis is pi/3 and the radius is 64.It is important to note that this represents the cubic version of z. We are interested in z. If the radius of z^3 is 64, then by taking its cubic root we can see that z has radius 4. This completes the first step of z= rexp(itheta + 2pi) = 4exp(itheta + 2pi)Now, we need to find the angle. Remember that the angle is measured from the positive x-axis going anticlockwise. We need to add pi to pi/3 to obtain 4pi/3. Remember that complex numbers repeat every 2pi! This leads to 4pi/3, 10pi/3, 16pi/3 Then arg(z^3) = 4pi/3; 10pi/3; 16pi/3 Either use the result that arg(z^N) = Narg(z) or use the properties of the exponential to see that if arg(z^3) = 4pi/3 then arg(x) = (arg(z^3)) / 3= (4pi/9). Z = 4exp(4pi/9). 4exp(10pi/9), 4xp(16pi/9).
