find the three cuberoots of 8( cos 264 + jsin 264) and state which is the principal cuberoot. Show all three roots on an argand diagram.
∛rcisθ = ∛r cis θ/3
so, we have to find all the angles between 0 and 360:
2 cis 88
2 cis (88+120)
2 cis (88+240)
To find the three cube roots of the given complex number, 8(cos 264 + jsin 264), we can follow these steps:
Step 1: Convert the given number to exponential form.
Using Euler's formula, we have:
8(cos 264 + jsin 264) = 8e^(j(264))
Step 2: Find the principal cube root.
The principal cube root of a complex number is the one with the smallest argument. To find the principal cube root, we divide the argument by 3:
Principal cube root = 8^(1/3) * e^(j(264/3))
Step 3: Find the other two cube roots.
To find the other two cube roots, we add multiples of 120 degrees (or 2*pi/3 radians) to the argument of the principal cube root.
Second cube root = 8^(1/3) * e^(j(264/3 + 2*pi/3))
Third cube root = 8^(1/3) * e^(j(264/3 + 4*pi/3))
Step 4: Express the cube roots in rectangular (Cartesian) form.
Using Euler's formula, we can convert the cube roots from exponential to rectangular form:
Principal cube root = 8^(1/3) * [cos(264/3) + jsin(264/3)]
Second cube root = 8^(1/3) * [cos(264/3 + 2*pi/3) + jsin(264/3 + 2*pi/3)]
Third cube root = 8^(1/3) * [cos(264/3 + 4*pi/3) + jsin(264/3 + 4*pi/3)]
Step 5: Plot the roots on an Argand diagram.
On the Argand diagram, the real part corresponds to the x-axis, and the imaginary part corresponds to the y-axis. Plot the rectangular form of each cube root on the diagram.
Note: To calculate the exact values of the cube roots, you can use a calculator with a complex number mode or write a program that handles complex numbers.