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DeMoivre Theorem

Grade 12 · Trigonometry · Worksheet 3

  1. Sophia is a marine biologist tracking the migration of a tagged shark. The shark's position relative to a research buoy is modeled by the complex number \( z = 1 - i \). Over the next several hours, the shark's movement follows a pattern that requires calculating \( z^6 \). Using De Moivre's Theorem, determine the exact rectangular form of \( (1 - i)^6 \). Answer: ______________
  2. Aroha is a Māori environmental scientist studying the seasonal growth pattern of a rare fern. The frond's spiral growth relative to the center of the plant is modeled by the complex number z = 9(cos(π/6) + i sin(π/6)). To predict the frond's exact position after 9 identical growth spurts, she needs to compute z^9. Using De Moivre's Theorem, determine the exact rectangular form of z^9. Answer: ______________
  3. Matiu is a Māori kaiako designing a digital animation of a traditional kōwhaiwhai pattern. The rotation of the pattern in the animation is governed by the complex number z = 4(cos(π/3) + i sin(π/3)). To complete a full cycle of the animation, the pattern must be rotated 18 times in succession, which corresponds to raising z to the 18th power. Using De Moivre's Theorem, determine the exact rectangular form of z^18. Answer: ______________
  4. A complex number is represented on the complex plane as a vector with magnitude 3 units and an angle of 120° from the positive real axis. Using De Moivre's Theorem, calculate the fourth power of this complex number. Express your answer in rectangular form (a + bi). Answer: ______________
  5. An electrical engineer is analyzing alternating current circuits and needs to calculate the voltage at a specific point in time. The voltage is given by the complex expression (√3 + i)^8. Using De Moivre's Theorem, determine the exact value of this voltage in rectangular form a + bi. Answer: ______________
  6. Noah is an aerospace engineer designing a satellite's communication system. The signal strength is modeled by the complex number z = 2(cos(π/7) + i sin(π/7)). To analyze the signal after it passes through 7 repeaters in sequence, he needs to compute z^14. Using De Moivre's Theorem, determine the exact rectangular form of z^14. Answer: ______________
  7. Use De Moivre's Theorem to compute (1 - i)^8 = ? Answer: ______________
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Answer Key & Explanations

DeMoivre Theorem · Grade 12 · Worksheet 3

  1. Sophia is a marine biologist tracking the migration of a tagged shark. The shark's position relative to a research buoy is modeled by the complex number \( z = 1 - i \). Over the next several hours, the shark's movement follows a pattern that requires calculating \( z^6 \). Using De Moivre's Theorem, determine the exact rectangular form of \( (1 - i)^6 \). Answer: 8i Solution: Convert 1 - i to polar form. Magnitude: r = sqrt(1^2 + (-1)^2) = sqrt(2). Angle: tan(theta) = -1/1 = -1.
    Full step-by-step solution

    Step 1: Convert 1 - i to polar form. Magnitude: r = sqrt(1^2 + (-1)^2) = sqrt(2). Angle: tan(theta) = -1/1 = -1. Since the point (1, -1) is in the fourth quadrant, theta = -pi/4 or 7pi/4. We use theta = -pi/4. So z = sqrt(2)[cos(-pi/4) + i sin(-pi/4)]. Step 2: Apply De Moivre's Theorem: z^6 = (sqrt(2))^6 [cos(6 * (-pi/4)) + i sin(6 * (-pi/4))]. (sqrt(2))^6 = (2^(1/2))^6 = 2^3 = 8. 6 * (-pi/4) = -6pi/4 = -3pi/2. Step 3: Evaluate cos(-3pi/2) and sin(-3pi/2). cos(-3pi/2) = 0. sin(-3pi/2) = 1. Step 4: Substitute back: z^6 = 8(0 + i * 1) = 8i. The answer is 8i.

  2. Aroha is a Māori environmental scientist studying the seasonal growth pattern of a rare fern. The frond's spiral growth relative to the center of the plant is modeled by the complex number z = 9(cos(π/6) + i sin(π/6)). To predict the frond's exact position after 9 identical growth spurts, she needs to compute z^9. Using De Moivre's Theorem, determine the exact rectangular form of z^9. Answer: -387420489i Solution: Identify r = 9 and θ = π/6. Apply De Moivre's Theorem: z^9 = [9(cos(π/6) + i sin(π/6))]^9 = 9^9 (cos(9 × π/6) + i sin(9 × π/6)). Simplify the angle: 9 × π/6 = 9π/6 = 3π/2.
    Full step-by-step solution

    Step 1: Identify r = 9 and θ = π/6. Step 2: Apply De Moivre's Theorem: z^9 = [9(cos(π/6) + i sin(π/6))]^9 = 9^9 (cos(9 × π/6) + i sin(9 × π/6)). Step 3: Simplify the angle: 9 × π/6 = 9π/6 = 3π/2. Step 4: Evaluate the trigonometric functions: cos(3π/2) = 0, sin(3π/2) = -1. Step 5: Substitute: z^9 = 9^9 (0 + i × (-1)) = 9^9 × (-i) = -9^9 i. Step 6: Compute 9^9: 9^2 = 81, 9^4 = 81^2 = 6561, 9^8 = 6561^2 = 43046721, 9^9 = 43046721 × 9 = 387420489. Therefore, z^9 = -387420489i in rectangular form.

  3. Matiu is a Māori kaiako designing a digital animation of a traditional kōwhaiwhai pattern. The rotation of the pattern in the animation is governed by the complex number z = 4(cos(π/3) + i sin(π/3)). To complete a full cycle of the animation, the pattern must be rotated 18 times in succession, which corresponds to raising z to the 18th power. Using De Moivre's Theorem, determine the exact rectangular form of z^18. Answer: 68719476736 Solution: Identify r = 4 and θ = π/3. Apply De Moivre's Theorem: z^18 = [4(cos(π/3) + i sin(π/3))]^18 = 4^18 (cos(18 × π/3) + i sin(18 × π/3)). Simplify the angle: 18 × π/3 = 6π.
    Full step-by-step solution

    Step 1: Identify r = 4 and θ = π/3. Step 2: Apply De Moivre's Theorem: z^18 = [4(cos(π/3) + i sin(π/3))]^18 = 4^18 (cos(18 × π/3) + i sin(18 × π/3)). Step 3: Simplify the angle: 18 × π/3 = 6π. Step 4: Evaluate the trigonometric functions: cos(6π) = 1, sin(6π) = 0 (since 6π is a multiple of 2π). Step 5: Substitute: z^18 = 4^18 (1 + i × 0) = 4^18. Step 6: Compute 4^18: 4^2 = 16, 4^4 = 16^2 = 256, 4^8 = 256^2 = 65536, 4^9 = 65536 × 4 = 262144, 4^18 = (4^9)^2 = 262144^2 = 68719476736. Therefore, z^18 = 68719476736 in rectangular form (since the imaginary part is 0). The answer is 68719476736.

  4. A complex number is represented on the complex plane as a vector with magnitude 3 units and an angle of 120° from the positive real axis. Using De Moivre's Theorem, calculate the fourth power of this complex number. Express your answer in rectangular form (a + bi). Answer: -81/2 + (81√3)/2 i Solution: Write the complex number in polar form: z = 3(cos 120° + i sin 120°) Apply De Moivre's Theorem: z^4 = 3^4 [cos(4 × 120°) + i sin(4 × 120°)] Calculate the magnitude: 3^4 = 81 Calculate the angle: 4 × 120° = 480° Find the equivalent angle between 0° and 360°: 480° - 360° = 120° Evaluate the…
    Full step-by-step solution

    Step 1: Write the complex number in polar form: z = 3(cos 120° + i sin 120°) Step 2: Apply De Moivre's Theorem: z^4 = 3^4 [cos(4 × 120°) + i sin(4 × 120°)] Step 3: Calculate the magnitude: 3^4 = 81 Step 4: Calculate the angle: 4 × 120° = 480° Step 5: Find the equivalent angle between 0° and 360°: 480° - 360° = 120° Step 6: Evaluate the trigonometric functions: cos 120° = -1/2, sin 120° = √3/2 Step 7: Multiply: z^4 = 81(-1/2 + i√3/2) = -81/2 + (81√3)/2 i The answer is -81/2 + (81√3)/2 i.

  5. An electrical engineer is analyzing alternating current circuits and needs to calculate the voltage at a specific point in time. The voltage is given by the complex expression (√3 + i)^8. Using De Moivre's Theorem, determine the exact value of this voltage in rectangular form a + bi. Answer: -128 + 128√3 i Solution: De Moivre's Theorem states that for any complex number in polar form r(cos θ + i sin θ) and any integer n, the nth power equals r^n(cos(nθ) + i sin(nθ)).
    Full step-by-step solution

    De Moivre's Theorem states that for any complex number in polar form r(cos θ + i sin θ) and any integer n, the nth power equals r^n(cos(nθ) + i sin(nθ)). This theorem is particularly useful for simplifying powers of complex numbers, as it avoids lengthy binomial expansions. In electrical engineering, this helps analyze AC circuits where voltages and currents are represented as complex numbers.

  6. Noah is an aerospace engineer designing a satellite's communication system. The signal strength is modeled by the complex number z = 2(cos(π/7) + i sin(π/7)). To analyze the signal after it passes through 7 repeaters in sequence, he needs to compute z^14. Using De Moivre's Theorem, determine the exact rectangular form of z^14. Answer: 16384 Solution: Identify the polar form: z = 2(cos(π/7) + i sin(π/7)) with r = 2 and θ = π/7. Apply De Moivre's Theorem: z^14 = [2(cos(π/7) + i sin(π/7))]^14 = 2^14 (cos(14 × π/7) + i sin(14 × π/7)). Simplify the angle: 14 × π/7 = 2π.
    Full step-by-step solution

    Step 1: Identify the polar form: z = 2(cos(π/7) + i sin(π/7)) with r = 2 and θ = π/7. Step 2: Apply De Moivre's Theorem: z^14 = [2(cos(π/7) + i sin(π/7))]^14 = 2^14 (cos(14 × π/7) + i sin(14 × π/7)). Step 3: Simplify the angle: 14 × π/7 = 2π. Step 4: Evaluate trigonometric functions: cos(2π) = 1, sin(2π) = 0. Step 5: Substitute: z^14 = 2^14 (1 + i × 0) = 2^14. Step 6: Calculate 2^14 = 16384. The answer is 16384.

  7. Use De Moivre's Theorem to compute (1 - i)^8 = ? Answer: 16 Solution: Convert 1 - i to polar form.
    Full step-by-step solution

    Step 1: Convert 1 - i to polar form. Modulus: r = sqrt(1^2 + (-1)^2) = sqrt(1 + 1) = sqrt(2) Argument: θ = arctan(-1/1) = -π/4 (since the point is in quadrant IV) So, 1 - i = sqrt(2)(cos(-π/4) + i sin(-π/4)) Step 2: Apply De Moivre's Theorem: (r(cosθ + i sinθ))^n = r^n(cos(nθ) + i sin(nθ)) Here, n = 8, r = sqrt(2), θ = -π/4 (1 - i)^8 = (sqrt(2))^8 (cos(8 × -π/4) + i sin(8 × -π/4)) Step 3: Calculate the modulus power: (sqrt(2))^8 = (2^(1/2))^8 = 2^(8/2) = 2^4 = 16 Step 4: Calculate the angle: 8 × (-π/4) = -2π Step 5: Evaluate trigonometric functions: cos(-2π) = cos(2π) = 1, sin(-2π) = -sin(2π) = 0 Step 6: Combine results: 16(1 + i × 0) = 16 The answer is 16.