Signals and Systems Theory Question

[OmniNewton]

Homework Statement

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How are we able to go from the first line to the second line and then the second line to the third?

Homework Equations

Euler Identity: e^j(theta) = cos(theta) +jsin(theta)

The Attempt at a Solution

This problem is more about preliminary theory in my opinion so I tried understanding how they converted the problem from trigonometric functions to exponential by analyzing the Euler Identity.

 

[rude man]

Rewrite sin(x) and cos(x) in terms of the Euler identity, substitute in the original equation, and force equivalences.

 

[OmniNewton]

Rewrite sin(x) and cos(x) in terms of the Euler identity, substitute in the original equation, and force equivalences.

Thank you for the response sir but I really do not see how that works. How can one simply say that 2.5cos(3t) = 2.5e^(3jt). I thought cos(theta) = 1/2(e^j(theta) + e^-j(theta)) determined by the subtraction of 2 mcclauirin series.

 

[rude man]

Thank you for the response sir but I really do not see how that works. How can one simply say that 2.5cos(3t) = 2.5e^(3jt).

You can't.

I thought cos(theta) = 1/2(e^j(theta) + e^-j(theta))

Right. Use that and the similar expression for sin(theta) and combine coefficients of e^j3t and e^-j3t.

 

[OmniNewton]

You can't. Right. Use that and the similar expression for sin(theta) and combine coefficients of e^j3t and e^-j3t.

Oh I see! That makes a lot of sense thank you kindly.