DeMoivre Theorem

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DeMoivre Theorem · Grade 12 · Worksheet 2

1. Matiu is a Māori carver designing a pattern for a traditional waka paddle. The carving tool's position in the design plane is modeled by the complex number z = 2(cos(π/4) + i sin(π/4)). To create a symmetrical repeating pattern, Matiu needs to apply the carving motion 8 times in succession, which corresponds to raising z to the 8th power. Using De Moivre's Theorem, find the exact rectangular form of z^8. Answer: 256 Solution: Identify the polar form: r = 2, θ = π/4. Apply De Moivre's Theorem: z^8 = [2(cos(π/4) + i sin(π/4))]^8 = 2^8 (cos(8 × π/4) + i sin(8 × π/4)). Simplify the magnitude: 2^8 = 256.
   Full step-by-step solution

   Step 1: Identify the polar form: r = 2, θ = π/4. Step 2: Apply De Moivre's Theorem: z^8 = [2(cos(π/4) + i sin(π/4))]^8 = 2^8 (cos(8 × π/4) + i sin(8 × π/4)). Step 3: Simplify the magnitude: 2^8 = 256. Step 4: Simplify the angle: 8 × π/4 = 2π. Step 5: Evaluate trigonometric functions: cos(2π) = 1, sin(2π) = 0. Step 6: Substitute back: z^8 = 256(1 + i × 0) = 256. The answer is 256.

2. A robotics engineer is programming a robotic arm that moves in a circular path. The arm's position is represented by the complex number z = 1 + i√3. To calculate the arm's position after completing 5 full rotations, she needs to find z^10. Using De Moivre's Theorem, determine the exact rectangular form of (1 + i√3)^10. Answer: -512 + 512i√3 Solution: Convert 1 + i√3 to polar form Find the modulus: r = sqrt(1^2 + (√3)^2) = sqrt(1 + 3) = sqrt(4) = 2 Find the argument: θ = arctan(√3/1) = arctan(√3) = π/3 So 1 + i√3 = 2(cos(π/3) + i sin(π/3)) (1 + i√3)^10 = [2(cos(π/3) + i sin(π/3))]^10 = 2^10 [cos(10π/3) + i sin(10π/3)] = 1024 [cos(10π/3) + i…
   Full step-by-step solution

   Step 1: Convert 1 + i√3 to polar form Find the modulus: r = sqrt(1^2 + (√3)^2) = sqrt(1 + 3) = sqrt(4) = 2 Find the argument: θ = arctan(√3/1) = arctan(√3) = π/3 So 1 + i√3 = 2(cos(π/3) + i sin(π/3)) Step 2: Apply De Moivre's Theorem (1 + i√3)^10 = [2(cos(π/3) + i sin(π/3))]^10 = 2^10 [cos(10π/3) + i sin(10π/3)] = 1024 [cos(10π/3) + i sin(10π/3)] Step 3: Simplify the angle 10π/3 = 3π + π/3 = π + 2π + π/3 Since cosine and sine are periodic with period 2π, we can subtract 2π: 10π/3 - 2π = 10π/3 - 6π/3 = 4π/3 So cos(10π/3) = cos(4π/3) and sin(10π/3) = sin(4π/3) Step 4: Evaluate trigonometric functions cos(4π/3) = cos(π + π/3) = -cos(π/3) = -1/2 sin(4π/3) = sin(π + π/3) = -sin(π/3) = -√3/2 Step 5: Multiply by the modulus 1024 [cos(4π/3) + i sin(4π/3)] = 1024 [-1/2 + i(-√3/2)] = 1024 × (-1/2) + 1024 × i(-√3/2) = -512 - 512i√3 The answer is -512 + 512i√3.

3. On the complex plane, Olivia draws a vector representing the complex number z. The vector has a magnitude of 5 units and makes an angle of 135° with the positive real axis. Using De Moivre's Theorem, find the fifth power of this complex number (z^5) and express your final answer in rectangular form a + bi. Answer: 3125sqrt(2)/2 * (-1 - i) or -3125sqrt(2)/2 - 3125sqrt(2)/2 i Solution: Write the complex number in polar form: z = 5(cos 135° + i sin 135°) Apply De Moivre's Theorem: z^5 = 5^5 [cos(5 × 135°) + i sin(5 × 135°)] Calculate the new magnitude: 5^5 = 3125 Calculate the new angle: 5 × 135° = 675° Reduce 675° to an equivalent angle between 0° and 360°: 675° - 360° = 315°…
   Full step-by-step solution

   Step 1: Write the complex number in polar form: z = 5(cos 135° + i sin 135°) Step 2: Apply De Moivre's Theorem: z^5 = 5^5 [cos(5 × 135°) + i sin(5 × 135°)] Step 3: Calculate the new magnitude: 5^5 = 3125 Step 4: Calculate the new angle: 5 × 135° = 675° Step 5: Reduce 675° to an equivalent angle between 0° and 360°: 675° - 360° = 315° Step 6: Write z^5 = 3125(cos 315° + i sin 315°) Step 7: Evaluate cos 315° and sin 315°. On the unit circle, 315° is in the fourth quadrant. The reference angle is 45°. cos 315° = cos 45° = sqrt(2)/2, but cosine is positive in QIV, so cos 315° = sqrt(2)/2. sin 315° = -sin 45° = -sqrt(2)/2. Step 8: Substitute: z^5 = 3125[ sqrt(2)/2 + i(-sqrt(2)/2) ] = 3125[ sqrt(2)/2 - sqrt(2)/2 i ] Step 9: Distribute: z^5 = 3125 sqrt(2)/2 - 3125 sqrt(2)/2 i The final answer is 3125sqrt(2)/2 - 3125sqrt(2)/2 i, which can also be written as (3125sqrt(2))/2 * (1 - i).

4. Find the fourth roots of 81(cos(7π/4) + i sin(7π/4)). Express your answers in rectangular form a + bi. Answer: 3√2/2 - 3√2/2 i, -3√2/2 + 3√2/2 i, -3√2/2 - 3√2/2 i, 3√2/2 + 3√2/2 i Solution: Identify parameters. r = 81, θ = 7π/4, n = 4. Modulus of roots: 81^(1/4) = 3.
   Full step-by-step solution

   Step 1: Identify parameters. r = 81, θ = 7π/4, n = 4. Step 2: Modulus of roots: 81^(1/4) = 3. Step 3: Apply formula for k = 0, 1, 2, 3: k = 0: θ₀ = (7π/4 + 0)/4 = 7π/16 k = 1: θ₁ = (7π/4 + 2π)/4 = (7π/4 + 8π/4)/4 = (15π/4)/4 = 15π/16 k = 2: θ₂ = (7π/4 + 4π)/4 = (7π/4 + 16π/4)/4 = (23π/4)/4 = 23π/16 k = 3: θ₃ = (7π/4 + 6π)/4 = (7π/4 + 24π/4)/4 = (31π/4)/4 = 31π/16 Step 4: Write roots in polar form: Root 1: 3(cos(7π/16) + i sin(7π/16)) Root 2: 3(cos(15π/16) + i sin(15π/16)) Root 3: 3(cos(23π/16) + i sin(23π/16)) Root 4: 3(cos(31π/16) + i sin(31π/16)) Step 5: Convert to rectangular form using exact values: cos(7π/16) = √(2-√2)/2, sin(7π/16) = √(2+√2)/2 → Root 1: 3(√(2-√2)/2 + i √(2+√2)/2) cos(15π/16) = -√(2+√2)/2, sin(15π/16) = √(2-√2)/2 → Root 2: 3(-√(2+√2)/2 + i √(2-√2)/2) cos(23π/16) = -√(2-√2)/2, sin(23π/16) = -√(2+√2)/2 → Root 3: 3(-√(2-√2)/2 - i √(2+√2)/2) cos(31π/16) = √(2+√2)/2, sin(31π/16) = -√(2-√2)/2 → Root 4: 3(√(2+√2)/2 - i √(2-√2)/2) Step 6: The four fourth roots are: 3√(2-√2)/2 + 3√(2+√2)/2 i, -3√(2+√2)/2 + 3√(2-√2)/2 i, -3√(2-√2)/2 - 3√(2+√2)/2 i, and 3√(2+√2)/2 - 3√(2-√2)/2 i.

5. On the complex plane, a vector representing a complex number z has a magnitude of 3 and makes an angle of 135° with the positive real axis. Using De Moivre's Theorem, find the three cube roots of this complex number. Express each root in rectangular form (a + bi) and describe their geometric arrangement on the complex plane. Answer: Cube roots: 3^(1/3)(cos 45° + i sin 45°) ≈ 1.1447(0.7071 + 0.7071i) ≈ 0.8090 + 0.8090i; 3^(1/3)(cos 165° + i sin 165°) ≈ 1.1447(-0.9659 + 0.2588i) ≈ -1.1056 + 0.2962i; 3^(1/3)(cos 285° + i sin 285°) ≈ 1.1447(0.2588 - 0.9659i) ≈ 0.2962 - 1.1056i. They are equally spaced at 120° intervals on a circle of radius cube root of 3. Solution: Write the complex number in polar form. Magnitude r = 3, angle θ = 135°. So z = 3(cos 135° + i sin 135°).
   Full step-by-step solution

   Step 1: Write the complex number in polar form. Magnitude r = 3, angle θ = 135°. So z = 3(cos 135° + i sin 135°). Step 2: To find cube roots, we use the formula for nth roots: z^(1/3) = 3^(1/3)[cos((135° + 360°k)/3) + i sin((135° + 360°k)/3)] for k = 0, 1, 2. Step 3: Calculate 3^(1/3). This is the cube root of 3, approximately 1.1447. Step 4: For k = 0: angle = (135° + 0°)/3 = 45°. Root 1 = 3^(1/3)(cos 45° + i sin 45°) = 1.1447(√2/2 + i√2/2) ≈ 1.1447(0.7071 + 0.7071i) = 0.8090 + 0.8090i. Step 5: For k = 1: angle = (135° + 360°)/3 = 495°/3 = 165°. Root 2 = 1.1447(cos 165° + i sin 165°). cos 165° = -cos 15° ≈ -0.9659, sin 165° = sin 15° ≈ 0.2588. So Root 2 ≈ 1.1447(-0.9659 + 0.2588i) = -1.1056 + 0.2962i. Step 6: For k = 2: angle = (135° + 720°)/3 = 855°/3 = 285°. Root 3 = 1.1447(cos 285° + i sin 285°). cos 285° = cos 75° ≈ 0.2588, sin 285° = -sin 75° ≈ -0.9659. So Root 3 ≈ 1.1447(0.2588 - 0.9659i) = 0.2962 - 1.1056i. Step 7: Geometric description: The three cube roots are equally spaced at 120° intervals (45°, 165°, 285°) on a circle centered at the origin with radius 3^(1/3) ≈ 1.1447. They form the vertices of an equilateral triangle. The cube roots are 0.8090 + 0.8090i, -1.1056 + 0.2962i, and 0.2962 - 1.1056i.

6. An electrical engineer is analyzing alternating current in a circuit with impedance represented by the complex number z = 2(cos(π/6) + i sin(π/6)). Using De Moivre's Theorem, determine the voltage amplitude when this impedance is raised to the 4th power in the circuit analysis. Answer: 16(cos(2π/3) + i sin(2π/3)) Solution: z = 2(cos(π/6) + i sin(π/6)) We want to compute z^4 using De Moivre's Theorem. If z = r(cos θ + i sin θ), then z^n = r^n (cos(nθ) + i sin(nθ)) for any positive integer n.
   Full step-by-step solution

   We are given the complex number: z = 2(cos(π/6) + i sin(π/6)) We want to compute z^4 using De Moivre's Theorem. --- **Step 1: Recall De Moivre's Theorem** De Moivre's Theorem states: If z = r(cos θ + i sin θ), then z^n = r^n (cos(nθ) + i sin(nθ)) for any positive integer n. --- **Step 2: Identify r and θ** From z = 2(cos(π/6) + i sin(π/6)), r = 2 θ = π/6 --- **Step 3: Apply De Moivre's Theorem for n = 4** z^4 = r^4 (cos(4θ) + i sin(4θ)) r^4 = 2^4 = 16 4θ = 4 × (π/6) = 4π/6 = 2π/3 So: z^4 = 16 (cos(2π/3) + i sin(2π/3)) --- **Step 4: Interpret the result** The problem asks for the voltage amplitude when impedance is raised to the 4th power. The amplitude is the modulus of z^4, which is 16. The full complex form is 16(cos(2π/3) + i sin(2π/3)). --- **Final Answer:** 16(cos(2π/3) + i sin(2π/3))

7. A robotics engineer is programming a robotic arm to trace a complex path in 3D space. The arm's position is represented by the complex number z = 1 + √3i, where the real part represents horizontal position and the imaginary part represents vertical position. To calculate the arm's position after 8 identical rotational movements, she needs to compute z⁸. Using De Moivre's Theorem, find the exact rectangular form of (1 + √3i)⁸. Answer: -128 - 128√3i Solution: Convert 1 + √3i to polar form Find the magnitude: r = √(1² + (√3)²) = √(1 + 3) = √4 = 2 Find the argument: θ = arctan(√3/1) = arctan(√3) = π/3 So 1 + √3i = 2(cos(π/3) + isin(π/3)) (1 + √3i)⁸ = [2(cos(π/3) + isin(π/3))]⁸ = 2⁸(cos(8×π/3) + isin(8×π/3)) 2⁸ = 256 8×π/3 = 8π/3 = (6π/3 + 2π/3) = 2π +…
   Full step-by-step solution

   Step 1: Convert 1 + √3i to polar form Find the magnitude: r = √(1² + (√3)²) = √(1 + 3) = √4 = 2 Find the argument: θ = arctan(√3/1) = arctan(√3) = π/3 So 1 + √3i = 2(cos(π/3) + isin(π/3)) Step 2: Apply De Moivre's Theorem (1 + √3i)⁸ = [2(cos(π/3) + isin(π/3))]⁸ = 2⁸(cos(8×π/3) + isin(8×π/3)) Step 3: Simplify 2⁸ = 256 8×π/3 = 8π/3 = (6π/3 + 2π/3) = 2π + 2π/3 Step 4: Use periodicity of trigonometric functions cos(2π + 2π/3) = cos(2π/3) = -1/2 sin(2π + 2π/3) = sin(2π/3) = √3/2 Step 5: Multiply 256 × (-1/2 + i√3/2) = 256 × (-1/2) + 256 × (i√3/2) = -128 + 128√3i Step 6: Write in standard form -128 + 128√3i The answer is -128 + 128√3i.