Find three cube roots for the following complex number. Leave your answers in trigonometric form. (Enter your answers from smallest to largest angle measure.)43 + 4i

Sure, here are the steps to find the cube roots of a complex number in trigonometric form:

Step 1: Convert the complex number to trigonometric form The trigonometric form of a complex number is r(cos θ + i sin θ), where r is the magnitude of the complex number and θ is the angle it makes with the positive x-axis.

The magnitude r is given by √(a² + b²), where a and b are the real and imaginary parts of the complex number respectively. So, r = √(43² + 4²) = √1850.

The angle θ is given by arctan(b/a), so θ = arctan(4/43) = 0.0934 radians.

So, the complex number in trigonometric form is √1850(cos 0.0934 + i sin 0.0934).

Step 2: Find the cube roots The cube roots of a complex number r(cos θ + i sin θ) are given by r^(1/3)[cos((θ+2πk)/3) + i sin((θ+2πk)/3)], where k = 0, 1, 2.

So, the cube roots are:

√1850^(1/3)[cos((0.0934+2π0)/3) + i sin((0.0934+2π0)/3)] √1850^(1/3)[cos((0.0934+2π1)/3) + i sin((0.0934+2π1)/3)] √1850^(1/3)[cos((0.0934+2π2)/3) + i sin((0.0934+2π2)/3)]

These are the three cube roots of the complex number 43 + 4i in trigonometric form.