solve for z writing the answer in polar form:
#1 z=(3,115 degrees)*(4,290 degrees)
#2 (8,50 degrees)* z=(6,172 degrees)
Multiply the magnitudes and add the angles to multiply
This comes from z = x + i y = r e^(i T)
here we have
3 e^i 115 * 4 e^i 290
You know to multiply 3*4 to get 12
now e^a * e^b = e^(a+b) (rules of exponents)
so we have 12 e^i(115+290) = 12 e^i 405
but 405 - 360 = 45
so we really have
modulus = 12
argument = 45 degrees
(6,172 degrees)
----------------
(8,50 degrees)
to divide, divide moduli, subtract angles (same discussion as for multiplication)
(.75) e^i (122)
by the way, the angles are generally expressed in radians for this notation.
To solve these equations, we need to use the properties of complex numbers in polar form. Complex numbers in polar form are expressed as z = r(cosθ + i sinθ), where r is the magnitude (or distance from the origin), and θ is the angle from the positive real axis (also known as the argument).
Let's solve the equations step by step:
#1 z = (3,115 degrees) * (4,290 degrees)
To multiply complex numbers in polar form, you can simply multiply their magnitudes and add their arguments:
Magnitude: 3 * 4 = 12.
Arguments: 115 + 290 = 405 degrees.
Therefore, the polar form of z is (12,405 degrees).
#2 (8,50 degrees) * z = (6,172 degrees)
To solve for z, we need to isolate it on one side of the equation. We can do this by dividing both sides of the equation by (8,50 degrees):
z = (6,172 degrees) / (8,50 degrees)
To divide complex numbers in polar form, divide their magnitudes and subtract their arguments:
Magnitude: 6 / 8 = 3/4.
Arguments: 172 - 50 = 122 degrees.
Therefore, the polar form of z is (3/4,122 degrees).