DeMoivre Theorem

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DeMoivre Theorem · Grade 12 · Worksheet 1

1. A robotics engineer is programming a robotic arm that moves in a complex plane. The arm's position is represented by the complex number z = 1 + i√3. To calculate the arm's position after performing 8 identical rotational movements, she needs to compute z^8. Using De Moivre's Theorem, what is the simplified rectangular form of (1 + i√3)^8? Answer: -128 - 128i√3 Solution: De Moivre's Theorem states that for a complex number in polar form r(cosθ + isinθ), raising it to the nth power gives r^n(cos(nθ) + isin(nθ)).
   Full step-by-step solution

   De Moivre's Theorem states that for a complex number in polar form r(cosθ + isinθ), raising it to the nth power gives r^n(cos(nθ) + isin(nθ)). This theorem is particularly useful for finding powers and roots of complex numbers, which has applications in engineering, physics, and signal processing. The key is to correctly identify the magnitude and argument of the complex number before applying the theorem.

2. Find the fifth roots of 16807(cos(9π/5) + i sin(9π/5)). Express your answers in rectangular form a + bi. Answer: 7(cos(9π/25) + i sin(9π/25)), 7(cos(19π/25) + i sin(19π/25)), 7(cos(29π/25) + i sin(29π/25)), 7(cos(39π/25) + i sin(39π/25)), 7(cos(49π/25) + i sin(49π/25)) Solution: Identify r = 16807, θ = 9π/5, n = 5. Magnitude of each root: r^(1/5) = 16807^(1/5) = 7 (since 7^5 = 16807). Angles: θ_k = (9π/5 + 2πk)/5 for k = 0, 1, 2, 3, 4.
   Full step-by-step solution

   Step 1: Identify r = 16807, θ = 9π/5, n = 5. Step 2: Magnitude of each root: r^(1/5) = 16807^(1/5) = 7 (since 7^5 = 16807). Step 3: Angles: θ_k = (9π/5 + 2πk)/5 for k = 0, 1, 2, 3, 4. k=0: (9π/5)/5 = 9π/25 k=1: (9π/5 + 2π)/5 = (9π/5 + 10π/5)/5 = (19π/5)/5 = 19π/25 k=2: (9π/5 + 4π)/5 = (9π/5 + 20π/5)/5 = (29π/5)/5 = 29π/25 k=3: (9π/5 + 6π)/5 = (9π/5 + 30π/5)/5 = (39π/5)/5 = 39π/25 k=4: (9π/5 + 8π)/5 = (9π/5 + 40π/5)/5 = (49π/5)/5 = 49π/25 Step 4: Polar forms: 7(cos(9π/25)+i sin(9π/25)), 7(cos(19π/25)+i sin(19π/25)), 7(cos(29π/25)+i sin(29π/25)), 7(cos(39π/25)+i sin(39π/25)), 7(cos(49π/25)+i sin(49π/25)). Step 5: These are the five fifth roots. If exact rectangular form is required, evaluate cos and sin of each angle using a calculator or known values. For example, cos(9π/25) ≈ 0.5358, sin(9π/25) ≈ 0.8443, so first root ≈ 7(0.5358 + 0.8443i) = 3.7506 + 5.9101i. Similar approximations can be made for the others. The five fifth roots are: 7(cos(9π/25)+i sin(9π/25)), 7(cos(19π/25)+i sin(19π/25)), 7(cos(29π/25)+i sin(29π/25)), 7(cos(39π/25)+i sin(39π/25)), 7(cos(49π/25)+i sin(49π/25)).

3. Ava is a quantum physicist studying the spin states of a particle in a magnetic field. The particle's state is represented by the complex number z = 7(cos(π/6) + i sin(π/6)). To predict the particle's behavior after 7 quantum transitions, she must compute z^7. Using De Moivre's Theorem, determine the exact rectangular form of z^7. Answer: 823543 * (sqrt(3)/2) - 823543/2 i Solution: Identify r = 7 and θ = π/6. Apply De Moivre's Theorem: z^7 = [7(cos(π/6) + i sin(π/6))]^7 = 7^7 (cos(7 × π/6) + i sin(7 × π/6)). Compute 7^7 = 7^3 × 7^4 = 343 × 2401 = 823543.
   Full step-by-step solution

   Step 1: Identify r = 7 and θ = π/6. Step 2: Apply De Moivre's Theorem: z^7 = [7(cos(π/6) + i sin(π/6))]^7 = 7^7 (cos(7 × π/6) + i sin(7 × π/6)). Step 3: Compute 7^7 = 7^3 × 7^4 = 343 × 2401 = 823543. Step 4: Compute the new angle: 7θ = 7 × π/6 = 7π/6. Step 5: Evaluate cos(7π/6): 7π/6 = π + π/6, so cos(7π/6) = -cos(π/6) = -sqrt(3)/2. Step 6: Evaluate sin(7π/6): sin(7π/6) = -sin(π/6) = -1/2. Step 7: Substitute: z^7 = 823543(-sqrt(3)/2 + i(-1/2)) = 823543(-sqrt(3)/2 - i/2). Step 8: Distribute 823543: z^7 = -823543*sqrt(3)/2 - 823543/2 i. Thus, the exact rectangular form is -823543*sqrt(3)/2 - 823543/2 i.

4. Emma is a telecommunications engineer designing a satellite communication system. The signal strength after passing through a specific amplifier is modeled by the complex number z = 3(cos(π/5) + i sin(π/5)). To analyze the signal after it has been amplified 5 times in sequence, she needs to compute z^5. Using De Moivre's Theorem, determine the exact rectangular form of z^5. Answer: -243 Solution: Identify r = 3 and θ = π/5. Apply De Moivre's Theorem: z^5 = [3(cos(π/5) + i sin(π/5))]^5 = 3^5 (cos(5 × π/5) + i sin(5 × π/5)). Compute 3^5 = 3 × 3 × 3 × 3 × 3 = 243.
   Full step-by-step solution

   Step 1: Identify r = 3 and θ = π/5. Step 2: Apply De Moivre's Theorem: z^5 = [3(cos(π/5) + i sin(π/5))]^5 = 3^5 (cos(5 × π/5) + i sin(5 × π/5)). Step 3: Compute 3^5 = 3 × 3 × 3 × 3 × 3 = 243. Step 4: Simplify the angle: 5 × π/5 = π. Step 5: Evaluate cos(π) = -1 and sin(π) = 0. Step 6: Substitute: z^5 = 243(-1 + i × 0) = 243 × (-1) = -243. Therefore, the exact rectangular form of z^5 is -243.

5. [2(cos 27° + i sin 27°)]⁵ = ? Answer: 32(cos 135° + i sin 135°) Solution: Identify r = 2, θ = 27°, n = 5. Apply De Moivre's Theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ). Compute rⁿ = 2⁵ = 32.
   Full step-by-step solution

   Step 1: Identify r = 2, θ = 27°, n = 5. Step 2: Apply De Moivre's Theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ). Step 3: Compute rⁿ = 2⁵ = 32. Step 4: Compute nθ = 5 × 27° = 135°. Step 5: Write the result: 32(cos 135° + i sin 135°). The answer is 32(cos 135° + i sin 135°).

6. Use De Moivre's Theorem to find the fourth roots of 16(cos(7π/12) + i sin(7π/12)). Express your answers in rectangular form a + bi. Answer: 2√2 + 2√2i, -2√2 + 2√2i, -2√2 - 2√2i, 2√2 - 2√2i Solution: Identify the parameters. We are finding the fourth roots (n = 4) of 16(cos(7π/12) + i sin(7π/12)). So r = 16 and θ = 7π/12.
   Full step-by-step solution

   Step 1: Identify the parameters. We are finding the fourth roots (n = 4) of 16(cos(7π/12) + i sin(7π/12)). So r = 16 and θ = 7π/12. Step 2: Apply De Moivre's Theorem for roots. The fourth roots are given by: 16^(1/4)[cos((7π/12 + 2πk)/4) + i sin((7π/12 + 2πk)/4)] for k = 0, 1, 2, 3. Step 3: Calculate the magnitude of the roots: 16^(1/4) = 2. Step 4: Calculate the angles for each root: For k = 0: θ₀ = (7π/12 + 0)/4 = 7π/48 For k = 1: θ₁ = (7π/12 + 2π)/4 = (7π/12 + 24π/12)/4 = (31π/12)/4 = 31π/48 For k = 2: θ₂ = (7π/12 + 4π)/4 = (7π/12 + 48π/12)/4 = (55π/12)/4 = 55π/48 For k = 3: θ₃ = (7π/12 + 6π)/4 = (7π/12 + 72π/12)/4 = (79π/12)/4 = 79π/48 Step 5: Write the roots in polar form: Root 1: 2(cos(7π/48) + i sin(7π/48)) Root 2: 2(cos(31π/48) + i sin(31π/48)) Root 3: 2(cos(55π/48) + i sin(55π/48)) Root 4: 2(cos(79π/48) + i sin(79π/48)) Step 6: Convert each root to rectangular form a + bi using exact values: Root 1: 2(cos(7π/48) + i sin(7π/48)) = 2(√2/2 + i √2/2) = √2 + √2i Root 2: 2(cos(31π/48) + i sin(31π/48)) = 2(-√2/2 + i √2/2) = -√2 + √2i Root 3: 2(cos(55π/48) + i sin(55π/48)) = 2(-√2/2 - i √2/2) = -√2 - √2i Root 4: 2(cos(79π/48) + i sin(79π/48)) = 2(√2/2 - i √2/2) = √2 - √2i Step 7: The four fourth roots are: √2 + √2i, -√2 + √2i, -√2 - √2i, and √2 - √2i.

7. Find the real part of the complex number (1 + i√3)^8 using De Moivre's Theorem. Express your answer as a simplified integer. Answer: -128 Solution: Write the complex number in polar form. We have z = 1 + i√3. r = sqrt( (1)^2 + (√3)^2 ) = sqrt(1 + 3) = sqrt(4) = 2.
   Full step-by-step solution

   Step 1: Write the complex number in polar form. We have z = 1 + i√3. First, find the modulus r: r = sqrt( (1)^2 + (√3)^2 ) = sqrt(1 + 3) = sqrt(4) = 2. Now, find the argument θ: tan θ = (√3)/1 = √3. Since the point is (1, √3) in the complex plane (real part 1, imaginary part √3), it is in the first quadrant. The angle whose tangent is √3 is π/3 (or 60 degrees). So θ = π/3. Thus, in polar form: z = 1 + i√3 = 2 (cos(π/3) + i sin(π/3)). 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/3. So z^8 = 2^8 [ cos(8 * π/3) + i sin(8 * π/3) ]. Step 3: Simplify the exponent. 2^8 = 256. Now compute 8 * π/3 = 8π/3. We need to reduce 8π/3 to an angle between 0 and 2π (or find its cosine directly). 8π/3 = (6π/3) + (2π/3) = 2π + 2π/3. Since cosine and sine are periodic with period 2π: cos(8π/3) = cos(2π + 2π/3) = cos(2π/3). sin(8π/3) = sin(2π + 2π/3) = sin(2π/3). Step 4: Evaluate the trigonometric functions. cos(2π/3) = cos(120°) = -1/2. sin(2π/3) = sin(120°) = √3/2. So z^8 = 256 [ (-1/2) + i (√3/2) ]. Step 5: Multiply to find the real part. z^8 = 256 * (-1/2) + i * 256 * (√3/2) = -128 + i * 128√3. The real part is -128. Final answer: -128

8. Use De Moivre's Theorem to 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 r = 81, θ = 7π/4, n = 4. Modulus of roots: 81^(1/4) = 3. For k = 0: angle = (7π/4 + 0)/4 = 7π/16.
   Full step-by-step solution

   Step 1: Identify r = 81, θ = 7π/4, n = 4. Step 2: Modulus of roots: 81^(1/4) = 3. Step 3: For k = 0: angle = (7π/4 + 0)/4 = 7π/16. Root: 3(cos(7π/16) + i sin(7π/16)). Step 4: For k = 1: angle = (7π/4 + 2π)/4 = (7π/4 + 8π/4)/4 = (15π/4)/4 = 15π/16. Root: 3(cos(15π/16) + i sin(15π/16)). Step 5: For k = 2: angle = (7π/4 + 4π)/4 = (7π/4 + 16π/4)/4 = (23π/4)/4 = 23π/16. Root: 3(cos(23π/16) + i sin(23π/16)). Step 6: For k = 3: angle = (7π/4 + 6π)/4 = (7π/4 + 24π/4)/4 = (31π/4)/4 = 31π/16. Root: 3(cos(31π/16) + i sin(31π/16)). Step 7: Convert each to rectangular form using exact values: cos(7π/16) = √(2-√2)/2, sin(7π/16) = √(2+√2)/2, so root 1 = 3√(2-√2)/2 + 3√(2+√2)/2 i. cos(15π/16) = -√(2+√2)/2, sin(15π/16) = √(2-√2)/2, so root 2 = -3√(2+√2)/2 + 3√(2-√2)/2 i. cos(23π/16) = -√(2-√2)/2, sin(23π/16) = -√(2+√2)/2, so root 3 = -3√(2-√2)/2 - 3√(2+√2)/2 i. cos(31π/16) = √(2+√2)/2, sin(31π/16) = -√(2-√2)/2, so root 4 = 3√(2+√2)/2 - 3√(2-√2)/2 i. 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.