for-what-value-of-e-i-0-

Question Number 61210 by pooja24 last updated on 30/May/19

f o r w h a t v a l u e o f θ, e^{i θ} = 0

Commented by Tony Lin last updated on 30/May/19

e^{i θ} = c o s θ + i s i n θ

c o s θ \in [- 1, 1] \in R

s i n θ \in [- 1, 1] \in R

\Rightarrow i f e^{i θ} = 0, t h e n s i n θ m u s t b e 0

\Rightarrow c o s θ m u s t b e 0

b u t {c o s}^{2} θ + {s i n}^{2} θ = 1

\Rightarrow e^{i θ} \neq 0

Answered by MJS last updated on 30/May/19

e^{i θ} usually is defined for - π ⩽ θ < π

r e^{i θ} = r \cos θ + i r \sin θ

in our case r = 1 \Rightarrow it can never be zero because

depending on θ it stands for all complex

numbers a + b i with r = \sqrt{a^{2} + b^{2}} = 1

Answered by aleks041103 last updated on 31/May/19

θ \to i \infty

\Rightarrow e^{i θ} = e^{- \infty} = 0