DeMoivre Theorem Worksheets Grade 12

1. Use De Moivre's Theorem to find the fourth roots of 81(cos(2π/3) + i sin(2π/3)). Express your answers in rectangular form a + bi.
2. Find the real part of the complex number (√3 + i)^6 using De Moivre's Theorem. Express your answer as a simplified integer.
3. Noah is a quantum optics researcher studying the behavior of a photon's polarization state in an optical fiber. The polarization state is modeled by the complex number z = 2(cos(π/6) + i sin(π/6)). After the photon passes through 6 identical polarizing filters in sequence, the resulting polarization state is given by z^6. Using De Moivre's Theorem, determine the exact rectangular form of z^6.

…and 5 more problems

1. 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.
2. A robotics engineer is programming a robotic arm that moves in a circular path. The arm's position is represented by the complex number z = 1 + i√3. To calculate the arm's position after completing 5 full rotations, she needs to find z^10. Using De Moivre's Theorem, determine the exact rectangular form of (1 + i√3)^10.
3. On the complex plane, Olivia draws a vector representing the complex number z. The vector has a magnitude of 5 units and makes an angle of 135° with the positive real axis. Using De Moivre's Theorem, find the fifth power of this complex number (z^5) and express your final answer in rectangular form a + bi.

…and 4 more problems

1. Use De Moivre's Theorem to find (2(cos(π/4) + i sin(π/4)))^8 = ?
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.
3. [2(cos 27° + i sin 27°)]⁵ = ?

…and 5 more problems