Why are these two complex exponentials equal?

Question

Why is $$e^{i(\theta+\pi)} = -e^{i\theta}$$ I saw this but I'm not sure why it works. If someone could show me the steps I would really appreciate it.

Answer 1

To show : $$e^{i(\theta+\pi)}=-e^{i\theta}$$

Now $$e^{i \theta} = \cos \theta + i \sin \theta \cdots (1)$$

so $$e^{i(\theta+\pi)}=\cos (\theta+\pi) + i \sin (\theta+\pi) \cdots (2)$$

Also , recall that $$\cos (\pi+\theta) = -\cos \theta $$ and $$\sin (\pi+\theta) = -\sin \theta $$

From the above results $(2)$ modifies to $$e^{i (\theta+\pi)} = -(\cos \theta + i \sin \theta)$$

Therefore from $(1)$ $$e^{i(\theta+\pi)}=-e^{i\theta}$$

Answer 2

Hint: this is wrong as it is, it should read $$e^{i(\theta+\pi)}=-e^{i\theta}$$ (theta and pi in the same brackets - edited).

Hint2: what functional equation does the exponential map satisfy? $e^{a+b}=\ldots$

Answer 3

Use $e^{i \theta} = \cos \theta + i \sin \theta$.