DeMoivre Theorem

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DeMoivre Theorem · Grade 12 · Worksheet 2

1. Find the real part of the complex number (√3 + i)^6 using De Moivre's Theorem. Express your answer as a simplified integer. Answer: -64 Solution: We have \( z = \sqrt{3} + i \). r = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 We know \( z = r(\cos\theta + i\sin\theta) \). Here, \( \cos\theta = \frac{\sqrt{3}}{2} \) and \( \sin\theta = \frac{1}{2} \).
   Full step-by-step solution

   Let's solve step-by-step using De Moivre's Theorem. --- **Step 1: Write the complex number in polar form** We have \( z = \sqrt{3} + i \). First, find the modulus \( r \): \[ r = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \] --- **Step 2: Find the argument \( \theta \)** We know \( z = r(\cos\theta + i\sin\theta) \). Here, \( \cos\theta = \frac{\sqrt{3}}{2} \) and \( \sin\theta = \frac{1}{2} \). From standard angles, \( \theta = \frac{\pi}{6} \) (or 30°). So: \[ z = 2\left( \cos\frac{\pi}{6} + i\sin\frac{\pi}{6} \right) \] --- **Step 3: Apply De Moivre's Theorem** De Moivre's Theorem says: \[ z^n = r^n \left( \cos(n\theta) + i\sin(n\theta) \right) \] Here \( n = 6 \), so: \[ z^6 = 2^6 \left( \cos\left(6 \cdot \frac{\pi}{6}\right) + i\sin\left(6 \cdot \frac{\pi}{6}\right) \right) \] \[ z^6 = 64 \left( \cos\pi + i\sin\pi \right) \] --- **Step 4: Evaluate cosine and sine** \[ \cos\pi = -1, \quad \sin\pi = 0 \] So: \[ z^6 = 64(-1 + i \cdot 0) = -64 \] --- **Step 5: Identify the real part** The number \( z^6 \) is purely real: \( -64 \). Thus, the real part is \( -64 \). --- **Final answer:** -64

2. 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.

3. 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.

4. (2(cos(π/4) + i sin(π/4)))^8 = ? Answer: 256 Solution: (2(cos(π/4) + i sin(π/4)))^8 2(cos(π/4) + i sin(π/4)) where r = 2 and θ = π/4. [r(cos θ + i sin θ)]^n = r^n (cos(nθ) + i sin(nθ)) Here, n = 8, r = 2, θ = π/4.
   Full step-by-step solution

   Let's solve step-by-step. We are given: (2(cos(π/4) + i sin(π/4)))^8 --- **Step 1: Recognize the complex number form** The expression inside the parentheses is: 2(cos(π/4) + i sin(π/4)) This is in polar form: r(cos θ + i sin θ) where r = 2 and θ = π/4. --- **Step 2: Apply De Moivre’s Theorem** De Moivre’s Theorem says: [r(cos θ + i sin θ)]^n = r^n (cos(nθ) + i sin(nθ)) Here, n = 8, r = 2, θ = π/4. So: (2(cos(π/4) + i sin(π/4)))^8 = 2^8 (cos(8 × π/4) + i sin(8 × π/4)) --- **Step 3: Compute the power of the modulus** 2^8 = 256. So we have: 256 (cos(8 × π/4) + i sin(8 × π/4)) --- **Step 4: Compute the new angle** 8 × π/4 = 8π/4 = 2π. So: cos(2π) = 1 sin(2π) = 0 --- **Step 5: Substitute back** 256 (1 + i × 0) = 256 × 1 = 256. --- **Final Answer:** 256

5. (3(cos(π/3) + i sin(π/3)))^4 = ? Answer: -81/2 + (81√3)/2 i Solution: Apply De Moivre's Theorem: (3(cos(π/3) + i sin(π/3)))^4 = 3^4 (cos(4π/3) + i sin(4π/3)) Calculate the modulus: 3^4 = 81 Calculate the angle: 4 × π/3 = 4π/3 Evaluate cos(4π/3) = -1/2 Evaluate sin(4π/3) = -√3/2 Substitute back: 81(-1/2 + i(-√3/2)) = -81/2 - (81√3)/2 i The answer is -81/2 - (81√3)/2 i.
   Full step-by-step solution

   Step 1: Apply De Moivre's Theorem: (3(cos(π/3) + i sin(π/3)))^4 = 3^4 (cos(4π/3) + i sin(4π/3)) Step 2: Calculate the modulus: 3^4 = 81 Step 3: Calculate the angle: 4 × π/3 = 4π/3 Step 4: Evaluate cos(4π/3) = -1/2 Step 5: Evaluate sin(4π/3) = -√3/2 Step 6: Substitute back: 81(-1/2 + i(-√3/2)) = -81/2 - (81√3)/2 i Step 7: The answer is -81/2 - (81√3)/2 i.

6. (3(cos(2π/3) + i sin(2π/3)))^4 = ? Answer: 81/2 - (81√3)/2 i Solution: Apply De Moivre's Theorem: (3(cos(2π/3) + i sin(2π/3)))^4 = 3^4 (cos(4 × 2π/3) + i sin(4 × 2π/3)) Calculate the modulus: 3^4 = 81 Calculate the new angle: 4 × 2π/3 = 8π/3 Simplify the angle: 8π/3 - 2π = 8π/3 - 6π/3 = 2π/3 The expression becomes: 81 (cos(2π/3) + i sin(2π/3)) Evaluate cos(2π/3) =…
   Full step-by-step solution

   Step 1: Apply De Moivre's Theorem: (3(cos(2π/3) + i sin(2π/3)))^4 = 3^4 (cos(4 × 2π/3) + i sin(4 × 2π/3)) Step 2: Calculate the modulus: 3^4 = 81 Step 3: Calculate the new angle: 4 × 2π/3 = 8π/3 Step 4: Simplify the angle: 8π/3 - 2π = 8π/3 - 6π/3 = 2π/3 Step 5: The expression becomes: 81 (cos(2π/3) + i sin(2π/3)) Step 6: Evaluate cos(2π/3) = -1/2 and sin(2π/3) = √3/2 Step 7: Substitute: 81 (-1/2 + i (√3/2)) = -81/2 + (81√3)/2 i The answer is -81/2 + (81√3)/2 i.

7. [7(cos(5π/6) + i sin(5π/6))]^4 = ? Answer: 2401(cos(10π/3) + i sin(10π/3)) or 2401(-1/2 - i√3/2) Solution: Identify r = 7, θ = 5π/6, n = 4. Apply De Moivre's Theorem: [r(cos θ + i sin θ)]^n = r^n(cos(nθ) + i sin(nθ)). Compute r^n = 7^4 = 2401.
   Full step-by-step solution

   Step 1: Identify r = 7, θ = 5π/6, n = 4. Step 2: Apply De Moivre's Theorem: [r(cos θ + i sin θ)]^n = r^n(cos(nθ) + i sin(nθ)). Step 3: Compute r^n = 7^4 = 2401. Step 4: Compute nθ = 4 × (5π/6) = 20π/6 = 10π/3. Step 5: The result in polar form is 2401(cos(10π/3) + i sin(10π/3)). Step 6: Simplify the angle: 10π/3 = 3π + π/3, so cos(10π/3) = cos(π + π/3) = -cos(π/3) = -1/2, and sin(10π/3) = sin(π + π/3) = -sin(π/3) = -√3/2. Step 7: In rectangular form: 2401(-1/2 - i√3/2) = -2401/2 - (2401√3/2)i. The answer is 2401(cos(10π/3) + i sin(10π/3)) or -2401/2 - (2401√3/2)i.

8. Use De Moivre's Theorem to find the fifth roots of 243(cos(5π/3) + i sin(5π/3)). Express your answers in rectangular form a + bi. Answer: 3/2 + (3√3/2)i, -3√3/2 + (3/2)i, -3 - 0i, -3√3/2 - (3/2)i, 3/2 - (3√3/2)i Solution: Identify r = 243, θ = 5π/3, n = 5. The modulus of each root is r^(1/5) = 243^(1/5) = 3. For k = 0: angle = (5π/3 + 0)/5 = 5π/15 = π/3.
   Full step-by-step solution

   Step 1: Identify r = 243, θ = 5π/3, n = 5. Step 2: The modulus of each root is r^(1/5) = 243^(1/5) = 3. Step 3: For k = 0: angle = (5π/3 + 0)/5 = 5π/15 = π/3. Root = 3(cos(π/3) + i sin(π/3)) = 3(1/2 + i√3/2) = 3/2 + (3√3/2)i. Step 4: For k = 1: angle = (5π/3 + 2π)/5 = (5π/3 + 6π/3)/5 = (11π/3)/5 = 11π/15. cos(11π/15) = -√3/2, sin(11π/15) = 1/2. Root = 3(-√3/2 + i/2) = -3√3/2 + (3/2)i. Step 5: For k = 2: angle = (5π/3 + 4π)/5 = (5π/3 + 12π/3)/5 = (17π/3)/5 = 17π/15. cos(17π/15) = -1, sin(17π/15) = 0. Root = 3(-1 + 0i) = -3. Step 6: For k = 3: angle = (5π/3 + 6π)/5 = (5π/3 + 18π/3)/5 = (23π/3)/5 = 23π/15. cos(23π/15) = -√3/2, sin(23π/15) = -1/2. Root = 3(-√3/2 - i/2) = -3√3/2 - (3/2)i. Step 7: For k = 4: angle = (5π/3 + 8π)/5 = (5π/3 + 24π/3)/5 = (29π/3)/5 = 29π/15. cos(29π/15) = 1/2, sin(29π/15) = -√3/2. Root = 3(1/2 - i√3/2) = 3/2 - (3√3/2)i. The five fifth roots are: 3/2 + (3√3/2)i, -3√3/2 + (3/2)i, -3, -3√3/2 - (3/2)i, 3/2 - (3√3/2)i.