Understanding Exponential Complex Numbers

[adartsesirhc]

Homework Statement

I've never understood $$e^{i\theta}$$ very well. I know that $$e^{i\theta} = cos \theta + i sin \theta$$, but how about $$e^{4i}$$? Would this be $$cos 1 + 4i sin 1$$ or $$cos 4 + i sin 4$$? What's the general rule for these kinds of numbers?

Homework Equations

$$e^{i\theta} = cos \theta + i sin \theta$$
$$e^{i\pi} + 1 = 0$$

The Attempt at a Solution

None, really. Just wondering how to evaluate numbers like above.

 

[snipez90]

I'm not sure where you got cos(1) + 4i*sin(1) from. For e^(4i), you would simply replace theta with 4, which gives the latter of the two expressions above. I assume we're dealing with radian measure so cos(4) + i*sin(4) can be evaluated using a calculator. On the complex plane, the point would be in the third quadrant since pi < 4 < (3/2)pi.

 

[HallsofIvy]

Homework Statement

I've never understood $$e^{i\theta}$$ very well. I know that $$e^{i\theta} = cos \theta + i sin \theta$$, but how about $$e^{4i}$$? Would this be $$cos 1 + 4i sin 1$$ or $$cos 4 + i sin 4$$? What's the general rule for these kinds of numbers?

The "rule" is given in the formula you state: $$e^{i\theta}= cos(\theta)+ i sin(\theta)$$. In [tex]e^{4i}[/itex], the number multiplying i is "4": $\theta= 4$. $$e^{4i}= cos(4)+ i sin(4)$$.<br /> <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> <h2>Homework Equations</h2><br /> $$e^{i\theta} = cos \theta + i sin \theta$$<br /> $$e^{i\pi} + 1 = 0$$<br /> <br /> <br /> <h2>The Attempt at a Solution</h2><br /> None, really. Just wondering how to evaluate numbers like above. </div> </div> </blockquote>[/tex]

 

[adartsesirhc]

Hmm... so does this mean that whatever you multiply $$i$$ by will be the argument of the sine and cosine? But my differential equations book has

$$e^{-ibx} = cos bx - i sin bx$$.

How do I know if this isn't

$$e^{-ibx} = e^{i(-bx)} = cos (-bx) + i sin (-bx)$$?

How do I tell when it's one and when it's the other?

 

[adartsesirhc]

OHHHH... never mind. =]

It just hit me: both are actually the same - just apply the appropriate trig identities.