# Thread: Complex number easy inequality

1. ## Complex number easy inequality

I was looking through a proof and i stumbled upon this inequality :

$\displaystyle | e^{ia}-1 | \leq |a |$

$\displaystyle a \in \mathbb{R}$

but i cannot convince myself why it is true.

Can you think about any hint?

thnx

2. I tried looking at it from this point but i am not sure how to involve $\displaystyle |a|$

$\displaystyle | e^{ia}-1 | = \sqrt{(cos(a) -1)^2 + sen^2(a) } = \sqrt{cos^2(a) - 2cos(a)+1 +sen^2(a)}$

3. Originally Posted by mabruka
I was looking through a proof and i stumbled upon this inequality :

$\displaystyle | e^{ia}-1 | \leq |a |$

$\displaystyle a \in \mathbb{R}$

but i cannot convince myself why it is true.

Can you think about any hint?

thnx
For a second suppose that $\displaystyle a\geqslant 0$ then $\displaystyle \left|e^{ia}-1\right|=\sqrt{\left(\cos(a)-1\right)^2+\sin^2(a)}=$$\displaystyle \sqrt{\cos^2(a)+\sin^2(a)+1-2\cos(a)}=\sqrt{2}\sqrt{1-\cos(a)}=2\sin\left(\frac{a}{2}\right)\leqslant 2\frac{a}{2}=a$

4. Thank you! its clear now