Final answer:
To convert between standard and trigonometric forms of complex numbers, we calculate the magnitude and argument for standard to trigonometric conversion, and apply the cosine and sine values for their respective angles in the reverse process.
Step-by-step explanation:
To convert complex numbers from standard form to trigonometric form and vice versa, we use polar coordinates and Euler's formula. For a given complex number in standard form z = a + bi, where a is the real part and b is the imaginary part, we can find the trigonometric form as z = r(cosθ + i sinθ), where r is the magnitude of z (given by r = √(a² + b²)) and θ is the argument of z (the angle formed by the line segment from the origin to the point (a, b) in the complex plane).
a) For z = 4 + 4i√3, we find the magnitude as r = √(4² + (4√3)²) = 8 and the argument θ using inverse trigonometric functions. Since the real and imaginary parts are positive, θ is in the first quadrant, θ = arctan(b/a), which yields θ = arctan(√3), so θ = 60°. Thus, the trigonometric form of z is z = 8(cos 60° + i sin 60°).
b) Converting z = 3(cos 60° + i sin 60°) into standard form requires using the cosine and sine values of the given angles. For 60°, the cosine value is 1/2, and the sine value is √3/2. Hence, the standard form is z = 3(1/2) + 3(√3/2)i or z = 3/2 + 3√3/2i.