Question

Find the fallowing in polar form

( 2 a z=1+i quad ) Write 2 in polar form b ( z¹⁰c find three cube root ( 2=1+i )

Answer 1

To convert the complex number 1+i to polar form, one would calculate the magnitude, √2, and the angle, 45 degrees, resulting in √2(cos(45°)+i sin(45°)). The three cube roots of 1+i are found by taking the cube root of the magnitude and dividing the angle by 3, then adding multiples of 120° for each root.

Step-by-step explanation:

Finding Polar Form and Cube Roots in Complex Numbers

The student's question seems to be asking about two separate problems in complex numbers:

1. Converting the complex number 1+i to polar form.
2. Finding the three cube roots of the given complex number 1+i.

To find the polar form of 1+i:

• Calculate the magnitude (r) using the formula √(x²+y²), which is √(1²+1²) = √2.
• Calculate the angle (θ) using the arctan(y/x), which is arctan(1/1) = π/4 or 45 degrees.

The polar form is r(cosθ+i sinθ) or √2(cos(45°)+i sin(45°)).

For the three cube roots:

• The magnitude of each root will be the cube root of √2.
• The angles will be θ/3 + k(360°/3) for k=0,1,2.

The three cube roots are then: (√2)¹³(cos(15°)+i sin(15°)), (√2)¹³(cos(135°)+i sin(135°)), and (√2)¹³(cos(255°)+i sin(255°)).