Polar Form (Complex Numbers — Advanced Applications)

Because √3 + i has modulus 2 and argument 30°, squaring it via De Moivre’s Theorem yields 4 cis 60°, which converts back into rectangular form as 4(cos 60° + i sin 60°) = 2 + √3 i, demonstrating how the theorem simplifies exponentiation of complex numbers.

Recognizing that 1 + i has modulus √2 and argument 45°, applying De Moivre’s Theorem gives (√2)⁴ cis(4×45°) = 4 cis 180°, which shows that raising the number to the fourth power quadruples the angle and squares the modulus twice to produce a final polar form of r = 4 and θ = 180°.

The product comes from multiplying moduli (3×2 = 6) and adding angles (30° + 60° = 90°), giving 6 cis 90°, representing a magnitude of 6 rotated exactly one quarter-turn counterclockwise.

By De Moivre's Theorem, squaring z multiplies the modulus by itself and doubles the angle: 3² = 9 and 2 × 120° = 240°. Result is 9 cis 240°. Option A doubles the modulus instead of squaring it, giving 6 instead of 9. Option C gives 180°, which has no derivation from 2 × 120°. Option D gives both wrong modulus and wrong angle.

The reference angle is tan⁻¹(4/3) ≈ 53.13°. Since the point (-3, 4) lies in quadrant II where x is negative and y is positive, the argument is 180° - 53.13° = 126.87°. Option A gives only the reference angle without the quadrant correction. Option B applies an incorrect adjustment of 90°. Option D incorrectly places the angle in quadrant III.

The cube roots of 1 are given by cis(0° + k·120°) for k = 0,1,2, reflecting the fact that unity sits at angle 0° and its roots lie evenly spaced every 120° on the unit circle, producing r = 1 at angles 0°, 120°, and 240°.

r = √((-3)² + 4²) = √(9 + 16) = √25 = 5. This follows the distance formula measuring how far the point (-3, 4) lies from the origin. Option A gives only the absolute value of the real part. Option B gives only the imaginary part. Option D adds 3 and 4 directly without squaring or taking a root.

Cubing the number multiplies its magnitude three times (giving 2³ = 8) and rotates its angle three times (45°×3 = 135°), resulting in 8 cis 135°, which reflects its new position in the second quadrant after triple rotation.

By dividing magnitudes (6 ÷ 2 = 3) and subtracting angles (120° − 30° = 90°), we obtain 3 cis 90°, illustrating how division corresponds to relative rotation and scaling between the two numbers.

The number -i corresponds to the point (0, -1), which lies one unit below the origin on the negative imaginary axis. Its modulus is r = 1 since the distance from the origin is 1. Its argument is θ = 270° because the negative imaginary axis is three-quarters of the way around the circle from the positive real axis. Option A gives 90° which is where +i lies on the positive imaginary axis, the opposite direction. Option B gives 180° which is the negative real axis, a quarter-turn short. Option D gives 0° which is the positive real axis.

Using Euler's formula: e^(iπ/2) = cos(π/2) + i sin(π/2) = 0 + i(1) = i. Multiplying by 5 gives 5i. The point lies 5 units up the positive imaginary axis. Option A gives a purely real result, ignoring that cos(π/2) = 0. Options B and C give negative values, contradicting sin(π/2) = +1.

By Euler's identity cosθ + i sinθ = e^(iθ), multiplying both sides by r gives r(cosθ + i sinθ) = r e^(iθ). Option A uses a negative exponent which produces the conjugate r(cosθ - i sinθ) instead. Option C omits the modulus r entirely. Option D doubles the angle with no mathematical justification.

Adding the angles (30° + 45° = 75°) and keeping the magnitude 1 gives cos 75° + i sin 75°, demonstrating the rotational nature of multiplying unit complex numbers.

Taking the cube root gives modulus 8^{1/3} = 2 and angles spaced evenly at 0° + k·120° for k = 0,1,2, producing roots 2 cis 0°, 2 cis 120°, and 2 cis 240°, forming an equilateral triangle on the complex plane.

Using cos 270° = 0 and sin 270° = −1 gives 8(0 + i·(−1)) = −8i, showing that the number points straight downward along the negative imaginary axis with magnitude 8.

Doubling the angle through De Moivre’s Theorem gives cos(240°) + i sin(240°), reflecting that squaring a point on the unit circle rotates it twice as far around the circle.

Using Euler’s formula e^{iθ} = cos θ + i sin θ converts e^{iπ/3} directly into cos 60° + i sin 60°, showing how the exponential representation elegantly encodes rotational motion.

Using De Moivre’s Theorem, raising a cis-number to the 5th power preserves its modulus (which is 1 here) and multiplies its angle by 5, giving cos(5θ) + i sin(5θ), illustrating how exponentiation corresponds to repeated rotation on the complex plane.

Dividing moduli (8 ÷ 2 = 4) and subtracting angles (120° − 60° = 60°) reflects how division in polar form corresponds to shrinking the magnitude while rotating backward, yielding a final expression of r = 4 and θ = 60°.

Multiplying these complex numbers requires multiplying their moduli (4×3 = 12) and adding their angles (60° + 30° = 90°), producing a new complex number whose polar representation is r = 12 and θ = 90°, consistent with the geometric interpretation of rotations and scalings.