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 e^{4i}[/itex], the number multiplying i is &quot;4&quot;: \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>

 

[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.