Exercises
Explore how trigonometry is used to represent and manipulate complex numbers. This quiz covers modulus, principal argument, trigonometric form, multiplication, division, powers, and nth roots. It also examines the geometric patterns formed by roots in the complex plane. The notation cis θ is used as shorthand for cos θ + i sin θ. Questions range from foundational conversions to applications of De Moivre's theorem.
Answer the questions below and check the explanation for each answer.
0/10 answered
Auto audio on: the next questions will be read aloud when you click Continue.
The modulus is |z| = √((-3)² + (3√3)²) = √(9 + 27) = √36 = 6.
The point lies in quadrant IV. Since tan θ = -√3, its reference angle is π/3, so the principal argument is -π/3.
The point is four units from the origin on the negative imaginary axis. Its modulus is 4 and a standard argument is 3π/2.
When complex numbers in trigonometric form are multiplied, their moduli multiply and their arguments add: 2 × 3 = 6 and π/6 + 3π/4 = 11π/12.
For division, divide the moduli and subtract the arguments. Thus 8 ÷ 2 = 4 and 5π/6 - π/3 = π/2, giving 4 cis(π/2).
De Moivre's theorem gives (√2)⁴ cis(4 × π/4) = 4 cis π. Since cis π = -1, the result is -4.
Cube roots have modulus ∛8 = 2. Their arguments are (0 + 2kπ)/3, so the roots are 2 cis(2kπ/3) for k = 0, 1, 2.
Every nonzero complex number has exactly n distinct nth roots. Therefore, z⁴ = 16 has four distinct complex solutions.
The cube roots are 1, cis(2π/3), and cis(4π/3). Their real and imaginary components cancel, so their sum is 0.
The roots have modulus √4 = 2 and arguments π/3 and π/3 + π = 4π/3. The angle 4π/3 lies in quadrant III, so the required root is 2 cis(4π/3).
Free CourseTrigonometry Grade 11
5
(1)
4h29m
39 exercises
Free CourseTrigonometry
4.6
(40)
6h35m
39 exercises
Free CourseTrigonometry
4.6
(17)
9h25m
32 exercises
Free CourseTrigonometry for beginners
4
(2)
45m
5 exercises
Free CourseTrigonometry full course
New
9h48m
6 exercises
Free CourseTrigonometry
New
16h32m
38 exercises
Free CourseTrigonometry Course
New
6h14m
52 exercises
Thousands of online courses in video, ebooks and audiobooks.
To test your knowledge during online courses
Generated directly from your cell phone's photo gallery and sent to your email
Download our app via QR Code or the links below:.
Carry knowledge in your pocket.
Download the Cursa app.
There are hundreds of free courses available, with a free certificate of completion that is saved in your mobile image gallery.
Download the app to access the Course Completion Certificate for Free.
+ 10 million
students
Free and Valid
Certificate
60 thousand free
exercises
4.8/5 rating in
app stores
Free courses in
video and ebooks