Complex Numbers Worksheets Grade 12

1. Liam is designing a drone navigation system that uses complex numbers to represent positions. His drone starts at the origin and flies to a point represented by the complex number -5 + 5√3 i. He needs to convert this position to polar form (r∠θ) where r is the distance from the origin and θ is the angle measured in degrees from the positive real axis, with -180° < θ ≤ 180°. What is the polar form of the drone's position?
2. Convert the complex number -4 - 4i to polar form r(cosθ + isinθ) = ?
3. Olivia is a sound engineer analyzing a complex audio signal. The signal's amplitude and phase at a certain frequency are represented by the complex number -5√2 + 5√2 i. To calibrate the sound system, she needs to express this signal in polar form r(cos θ + i sin θ), where r is the magnitude and θ is the phase angle in degrees between 0° and 360°. What is the polar form of this complex signal?

…and 5 more problems

1. On the complex plane, Matiu observes a point representing a complex number located at coordinates (-7, 7√3). Express this complex number in polar form r(cos θ + i sin θ), where r > 0 and θ is the principal argument in radians between 0 and 2π.
2. Convert the complex number 3 + 4i to polar form (r(cosθ + i sinθ)) = ?
3. Noah is analyzing a mechanical vibration system where the displacement of a spring-mass system is modeled by the complex number -7 + 7√3 i millimeters. To determine the amplitude and phase angle of the oscillation, he must convert this displacement to polar form r(cos θ + i sin θ) with θ in radians between 0 and 2π. What is the polar representation of the displacement?

…and 4 more problems

1. Liam is designing a drone navigation system that uses complex numbers to represent positions. His drone starts at the origin and flies to a point represented by the complex number -5 + 12i. To program the return flight path, he needs to convert this position to polar form (r∠θ), where r is the distance from the origin and θ is the angle measured in degrees from the positive real axis, with -180° < θ ≤ 180°. What is the polar form of the drone's position?
2. Liam is designing a drone navigation system that uses complex numbers to represent positions. His drone needs to fly from its current position at 2 + 2i to a target position at -1 + 3i. To calculate the flight vector, he needs to find the difference between these complex numbers and express it in polar form (r∠θ) where r is the distance and θ is the angle measured in degrees from the positive real axis. What is the polar form of the flight vector?
3. Mason is an audio engineer analyzing a sound wave captured by a microphone. The wave is represented by the complex number -8 - 8√3 i in the complex plane, where the real axis represents the in-phase component and the imaginary axis represents the quadrature component. To determine the amplitude and phase shift of the wave, Mason needs to convert this complex number to polar form r(cos θ + i sin θ), where r is the amplitude and θ is the phase angle in degrees between -180° and 180°. What is the polar representation of the sound wave?

…and 4 more problems