Polar Form (Complex Numbers — Advanced Applications)

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°.

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.

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.

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°.

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.

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°.

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.

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.

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 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.

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 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.

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.

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.

Using Euler’s identity cos θ + i sin θ = e^{iθ}, the expression becomes r e^{iθ}, highlighting the elegant connection between trigonometric and exponential representations of complex numbers.

Since cos(π/2) = 0 and sin(π/2) = 1, multiplying gives 5(0 + i·1) = 5i, placing the point five units above the origin on the imaginary axis.

Using r = √(x² + y²) gives √((-3)² + 4²) = √(9 + 16) = 5, which represents the distance from the origin to the point (−3, 4).

Computing tan θ = 4/−3 and recognizing that the point lies in Quadrant II gives θ = 180° − arctan(4/3) ≈ 126.87°, balancing both direction and magnitude of the rotation.

Squaring z produces modulus 3² = 9 and angle 2×120° = 240°, giving 9 cis 240°, reflecting a doubling of rotation with squared magnitude.

Since −i lies one unit below the origin on the negative imaginary axis, its modulus is 1 and its argument is 270°, forming the polar representation r = 1, θ = 270°.