به انگلیسی

Angles that multiply

Mathematics · Algebra · ages 17-18

Name ______________________ Date ____________

What is the real part of 6e^{i pi/3}?
Answer: ______________
Which is 2e^{i pi} in Cartesian form?
- 2
- 0
- 2i
- -2
What is e to the power i times 3pi/2?
- i
- -1
- -i
To divide in exponential form, Sam divides the moduli and subtracts the arguments.
Circle one: True False
What is the modulus of (10e^{i pi/2}) divided by (2e^{i pi/3})?
Answer: ______________
What is (cos 40 degrees + i sin 40 degrees) squared?
- cos 80 degrees + i sin 80 degrees
- cos 20 degrees + i sin 20 degrees
- Double the whole bracket
What is 3e^{i pi/2} times 2e^{i pi/2}, written in Cartesian form?
- 6i
- -6
- 6
- 5e^{i pi}
Ana says e^{i 2pi/3} equals e^{-i 4pi/3}. Is Ana right?
Circle one: True False
What is (cos 25 degrees + i sin 25 degrees) cubed?
- cos 15625 degrees + i sin 15625 degrees
- Triple the whole bracket
- cos 75 degrees + i sin 75 degrees
What is (cos 10 degrees + i sin 10 degrees) to the power 9?
- cos 19 degrees + i sin 19 degrees
- 9(cos 10 degrees + i sin 10 degrees)
- i
- -1

Answer key

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

3 · The real part is r cos T = 6 cos(pi/3) = 6 times 0.5 = 3.
-2 · e^{i pi} is -1, so 2e^{i pi} is -2.
-i · Euler at angle 3pi/2 gives cos 3pi/2 + i sin 3pi/2 = 0 - i = -i.
True · Division mirrors multiplication: divide the moduli and subtract the arguments.
5 · Divide the moduli: 10/2 = 5.
cos 80 degrees + i sin 80 degrees · De Moivre doubles the angle while modulus 1 squared stays 1.
-6 · Moduli multiply to 6 and arguments add to pi, so the product is 6e^{i pi} = -6.
True · The two angles differ by 2pi, a full turn, so they point the same way on the unit circle.
cos 75 degrees + i sin 75 degrees · Cubing triples the angle to 75 degrees and keeps modulus 1.
i · De Moivre: the angle becomes 9 times 10 = 90 degrees with modulus 1, and cos 90 + i sin 90 = i.