Angles that multiply

Mathematics · Algebra · ages 17-18

Name ______________________ Date ____________

What is the modulus of e^{2i}?
Answer: ______________
What is e to the power i times pi?
- 1
- 0
- i
- -1
What is e to the power i times pi?
- 1
- i
- -1
To multiply, Sam multiplies the moduli and adds the arguments.
Circle one: True False
What is the modulus of (2e^{i pi/4}) divided by (4e^{i pi/4})?
Answer: ______________
What is (cos 30 degrees + i sin 30 degrees) squared?
- cos 60 degrees + i sin 60 degrees
- cos 15 degrees + i sin 15 degrees
- Double the whole bracket
What is 2e^{i pi/6} times 3e^{i pi/3}, written in Cartesian form?
- 6
- 6i
- -6i
- 5e^{i pi/2}
Ana says e^{i pi/2} equals i. Is Ana right?
Circle one: True False
What is (cos 20 degrees + i sin 20 degrees) cubed?
- cos 8000 degrees + i sin 8000 degrees
- Triple the whole bracket
- cos 60 degrees + i sin 60 degrees
What is (cos 20 degrees + i sin 20 degrees) cubed?
- cos 8000 degrees + i sin 8000 degrees
- 3(cos 20 degrees + i sin 20 degrees)
- cos 60 degrees + i sin 60 degrees
- cos 7 degrees + i sin 7 degrees

Answer key

For grown-ups. Fold this page away before handing over the rest.

1 · 1. Every point e^{iT} sits on the unit circle, whatever T is.
-1 · -1. Euler with angle pi gives cos pi + i sin pi = -1 + 0i.
-1 · Euler at angle pi gives cos pi + i sin pi = -1 + 0i.
True · Moduli multiply and arguments add: that is the point of exponential form.
0.5 · 0.5. The arguments cancel and 2/4 = 0.5.
cos 60 degrees + i sin 60 degrees · De Moivre doubles the angle while modulus 1 squared stays 1.
6i · 6i. The product is 6e^{i pi/2}, and e^{i pi/2} = i.
True · Ana is right. cos(pi/2) = 0 and sin(pi/2) = 1, giving 0 + 1i.
cos 60 degrees + i sin 60 degrees · Cubing triples the angle to 60 degrees and keeps modulus 1.
cos 60 degrees + i sin 60 degrees · cos 60 degrees + i sin 60 degrees. Cubing triples the angle and keeps modulus 1.