DeMoivre Theorem

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DeMoivre Theorem · Grade 12 · Worksheet 3

1. Use De Moivre's Theorem to find (2(cos(π/4) + i sin(π/4)))^8 = ? Answer: 256 Solution: We are given: (2(cos(π/4) + i sin(π/4)))^8 Identify the modulus and argument. The complex number is in polar form: r(cos θ + i sin θ), where r = 2 and θ = π/4. Apply De Moivre's Theorem.
   Full step-by-step solution

   We are given: (2(cos(π/4) + i sin(π/4)))^8 Step 1: Identify the modulus and argument. The complex number 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. Step 3: Compute r^n. r^n = 2^8 = 256. Step 4: Compute nθ. nθ = 8 × (π/4) = 8π/4 = 2π. Step 5: Substitute into the formula. (2(cos(π/4) + i sin(π/4)))^8 = 256 (cos(2π) + i sin(2π)). Step 6: Evaluate cos(2π) and sin(2π). cos(2π) = 1, sin(2π) = 0. Step 7: Write the final result. 256 (1 + i × 0) = 256. So the answer is 256.

2. On the complex plane, a vector representing the complex number z has a magnitude of 6 units and makes an angle of 216° measured counterclockwise from the positive real axis. Using De Moivre's Theorem, find the value of z raised to the 5th power. Express your final answer in rectangular form a + bi, and visualize the resulting angle in standard position. Answer: 7776 + 0i Solution: Write the complex number in polar form: z = 6(cos 216° + i sin 216°). Apply De Moivre's Theorem: z^5 = 6^5 [cos(5 × 216°) + i sin(5 × 216°)]. Calculate the magnitude: 6^5 = 7776.
   Full step-by-step solution

   Step 1: Write the complex number in polar form: z = 6(cos 216° + i sin 216°). Step 2: Apply De Moivre's Theorem: z^5 = 6^5 [cos(5 × 216°) + i sin(5 × 216°)]. Step 3: Calculate the magnitude: 6^5 = 7776. Step 4: Calculate the angle: 5 × 216° = 1080°. Step 5: Reduce 1080° to a coterminal angle between 0° and 360°: 1080° - 3(360°) = 1080° - 1080° = 0°. Step 6: Evaluate the trigonometric functions: cos 0° = 1, sin 0° = 0. Step 7: Write the result in rectangular form: z^5 = 7776(1 + 0i) = 7776 + 0i. The answer is 7776 + 0i.

3. [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°).

4. [3(cos(5π/9) + i sin(5π/9))]⁶ = ? Answer: 729(cos(10π/3) + i sin(10π/3)) = 729(-1/2 - i√3/2) Solution: Identify r = 3 and θ = 5π/9. Apply De Moivre's Theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ). Here n = 6.
   Full step-by-step solution

   Step 1: Identify r = 3 and θ = 5π/9. Step 2: Apply De Moivre's Theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ). Here n = 6. Step 3: Compute rⁿ = 3⁶ = 729. Step 4: Compute nθ = 6 × (5π/9) = 30π/9 = 10π/3. Step 5: The result in polar form is 729(cos(10π/3) + i sin(10π/3)). Step 6: Simplify the angle: 10π/3 = 3π + π/3. Since cos and sin have period 2π, cos(10π/3) = cos(10π/3 - 2π) = cos(4π/3). Similarly, sin(10π/3) = sin(4π/3). Step 7: cos(4π/3) = -1/2, sin(4π/3) = -√3/2. Step 8: The final answer is 729(-1/2 - i√3/2).

5. Sophia is a quantum physicist analyzing the spin state of a particle in a magnetic field. The particle's state is represented by the complex number z = 7(cos(π/5) + i sin(π/5)). To predict the particle's behavior after 10 identical time evolution steps, she needs to compute z^10. Using De Moivre's Theorem, determine the exact rectangular form of z^10. Answer: 282475249 Solution: Identify r = 7 and θ = π/5. Apply De Moivre's Theorem: z^10 = [7(cos(π/5) + i sin(π/5))]^10 = 7^10 (cos(10 × π/5) + i sin(10 × π/5)). Simplify the angle: 10 × π/5 = 2π.
   Full step-by-step solution

   Step 1: Identify r = 7 and θ = π/5. Step 2: Apply De Moivre's Theorem: z^10 = [7(cos(π/5) + i sin(π/5))]^10 = 7^10 (cos(10 × π/5) + i sin(10 × π/5)). Step 3: Simplify the angle: 10 × π/5 = 2π. Step 4: Evaluate the trigonometric functions: cos(2π) = 1, sin(2π) = 0. Step 5: Substitute: z^10 = 7^10 (1 + i × 0) = 7^10. Step 6: Compute 7^10: 7^2 = 49, 7^4 = 49^2 = 2401, 7^5 = 2401 × 7 = 16807, 7^10 = 16807^2 = 282475249. Therefore, z^10 = 282475249 in rectangular form (since the imaginary part is 0). The answer is 282475249.

6. Mason is an electrical engineer designing a power transmission system. The voltage in a section of the circuit is represented by the complex number z = 9(cos(2π/9) + i sin(2π/9)). To analyze harmonic distortion, he needs to compute z^9. Using De Moivre's Theorem, determine the exact rectangular form of z^9. Answer: 387420489 Solution: Write the complex number in polar form: z = 9(cos(2π/9) + i sin(2π/9)). Apply De Moivre's Theorem: z^9 = [9(cos(2π/9) + i sin(2π/9))]^9 = 9^9 (cos(9 × 2π/9) + i sin(9 × 2π/9)). Simplify the angle: 9 × 2π/9 = 2π.
   Full step-by-step solution

   Step 1: Write the complex number in polar form: z = 9(cos(2π/9) + i sin(2π/9)). Step 2: Apply De Moivre's Theorem: z^9 = [9(cos(2π/9) + i sin(2π/9))]^9 = 9^9 (cos(9 × 2π/9) + i sin(9 × 2π/9)). Step 3: Simplify the angle: 9 × 2π/9 = 2π. Step 4: Evaluate the trigonometric functions: cos(2π) = 1, sin(2π) = 0. Step 5: Substitute back: z^9 = 9^9 (1 + i × 0) = 9^9. Step 6: Calculate 9^9: 9^2 = 81, 9^4 = 81^2 = 6561, 9^8 = 6561^2 = 43046721, then 9^9 = 9^8 × 9 = 43046721 × 9 = 387420489. Therefore, z^9 = 387420489 in rectangular form (since the imaginary part is 0). The answer is 387420489.

7. (√3 + i)^6 = ? Answer: -64 Solution: Convert √3 + i to polar form r = √((√3)² + 1²) = √(3 + 1) = √4 = 2 θ = arctan(1/√3) = π/6 So √3 + i = 2(cos(π/6) + i sin(π/6)) (2(cos(π/6) + i sin(π/6)))^6 = 2^6(cos(6 × π/6) + i sin(6 × π/6)) = 64(cos(π) + i sin(π)) cos(π) = -1 sin(π) = 0 So 64(-1 + i × 0) = 64 × -1 = -64 The answer is -64.
   Full step-by-step solution

   Step 1: Convert √3 + i to polar form r = √((√3)² + 1²) = √(3 + 1) = √4 = 2 θ = arctan(1/√3) = π/6 So √3 + i = 2(cos(π/6) + i sin(π/6)) Step 2: Apply De Moivre's Theorem (2(cos(π/6) + i sin(π/6)))^6 = 2^6(cos(6 × π/6) + i sin(6 × π/6)) = 64(cos(π) + i sin(π)) Step 3: Evaluate the trigonometric functions cos(π) = -1 sin(π) = 0 So 64(-1 + i × 0) = 64 × -1 = -64 The answer is -64.

8. An electrical engineer is designing a circuit that requires calculating the voltage across a component using complex impedance. The impedance is given by Z = 3(cos(π/6) + i sin(π/6)) ohms, and the current flowing through it is I = 2(cos(π/4) + i sin(π/4)) amperes. Using De Moivre's theorem, find the voltage V = Z × I in polar form, where the magnitude is in volts and the angle is in radians. Answer: 6(cos(5π/12) + i sin(5π/12)) Solution: Write down the given impedance and current in polar form. Impedance: Z = 3(cos(π/6) + i sin(π/6)) Current: I = 2(cos(π/4) + i sin(π/4)) If z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), then z1 × z2 = r1 × r2 [cos(θ1 + θ2) + i sin(θ1 + θ2)].
   Full step-by-step solution

   Step 1: Write down the given impedance and current in polar form. Impedance: Z = 3(cos(π/6) + i sin(π/6)) Current: I = 2(cos(π/4) + i sin(π/4)) Step 2: Recall that for multiplication in polar form: If z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), then z1 × z2 = r1 × r2 [cos(θ1 + θ2) + i sin(θ1 + θ2)]. Step 3: Identify the magnitudes and angles from Z and I. Magnitude of Z: r1 = 3 Angle of Z: θ1 = π/6 Magnitude of I: r2 = 2 Angle of I: θ2 = π/4 Step 4: Multiply the magnitudes. r1 × r2 = 3 × 2 = 6 Step 5: Add the angles. θ1 + θ2 = π/6 + π/4 Find a common denominator: π/6 = 2π/12 π/4 = 3π/12 Sum: 2π/12 + 3π/12 = 5π/12 Step 6: Write the result in polar form. V = Z × I = 6 [cos(5π/12) + i sin(5π/12)] Final answer: 6(cos(5π/12) + i sin(5π/12)) volts.