DeMoivre Theorem Worksheets Grade 12

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?
2. Find the fifth roots of 16807(cos(9π/5) + i sin(9π/5)). Express your answers in rectangular form a + bi.
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.

…and 5 more problems

1. Find the real part of the complex number (√3 + i)^6 using De Moivre's Theorem. Express your answer as a simplified integer.
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.
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.

…and 5 more problems

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 \( (1 - i)^6 \).
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.
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.

…and 4 more problems