$\cos θ + i \sin θ$ does not equal 0

By Arun Iyer (P4587) on Wednesday, June 13, 2001 - 09:27 pm :

Can you prove that

$\cos θ + i \sin θ$ does not equal 0 for any value of $θ$

By Dan Goodman (Dfmg2) on Wednesday, June 13, 2001 - 09:40 pm :

Arun, do you know that

\cos^{2} θ + \sin^{2} θ = 1

? If so, here is a proof.

If $\cos θ + i \sin θ = 0$ then also $\cos θ - i \sin θ = 0$ , so $(\cos θ + i \sin θ) (\cos θ - i \sin θ) = 0$ .

Expanding out the brackets on the left hand side you would get $\cos^{2} θ + \sin^{2} θ = 0$ , which is a contradiction because it should be 1.

Alternatively, just solve the simultaneous equations (1) $\cos θ = 0$ , (2) $\sin θ = 0$ . There are no solutions so we're OK.

By Jordan Dimov (M1178) on Thursday, June 14, 2001 - 06:00 pm :

Arun,

There is a sweet little formula due to Euler, which states that cos(x)+isin(x)=e^ix . Now, regardless of what x is, e^ix can never be zero (you do recall that e⁰ =1). This proves your statement.

-- Jordan

By Arun Iyer (P4587) on Thursday, June 14, 2001 - 08:01 pm :

Thanks Dan and Jordan

Arun