Ugly trig and exponential problem

[gulfcoastfella]

Homework Statement

This isn't a problem, I'm just obsessed with analyzing a trig/exp equation algebraically instead of with a calculator.

8*sin(t) - 16*cos(t) = 9*exp(-t/2)

Homework Equations

See part (1.) above...

The Attempt at a Solution

I tried converting the exp part into a Fourier series; if you graph the equation, this method recommends itself due to the multiple solutions. I didn't have much success with this method, though, since the first term in the Fourier series contains two exp terms. In addition, the Fourier series requires the definition of an integrable "period", and I wouldn't know what to do for that.

I also tried converting the sine and cosine terms into exp terms with complex arguments, but then I've got imaginary numbers all over the place, and I already know the answer isn't complex from the solution obtained by calculator, which was verified graphically.

Any recommendations would be grand.

 

[gulfcoastfella]

Homework Statement

This isn't a problem, I'm just obsessed with analyzing a trig/exp equation algebraically instead of with a calculator.

8*sin(t) - 16*cos(t) = 9*exp(-t/2)

Homework Equations

See part (1.) above...

The Attempt at a Solution

I tried converting the exp part into a Fourier series; if you graph the equation, this method recommends itself due to the multiple solutions. I didn't have much success with this method, though, since the first term in the Fourier series contains two exp terms. In addition, the Fourier series requires the definition of an integrable "period", and I wouldn't know what to do for that.

I also tried converting the sine and cosine terms into exp terms with complex arguments, but then I've got imaginary numbers all over the place, and I already know the answer isn't complex from the solution obtained by calculator, which was verified graphically.

Any recommendations would be grand.

I forgot to mention another method I attempted...
I converted both sides of the equation (the trig side and the exp side) to Taylor series. The terms line up beautifully, but the constants multiplied on each term in the original equation throw the equations out of whack.

 

[gulfcoastfella]

I've decided that this equation is so far past non-linear that it can't be solved algebraically. I'm going to go with the graphical intersection method, and let the problem go.

 

[HallsofIvy]

Did you try writing sin(x) as $(e^{ix}- e^{-ix})/(2i)$ and cos(x) as $(e^{ix}+ e^{-ix})/2$ so everything is in terms of the exponential?

 

[gulfcoastfella]

Did you try writing sin(x) as $(e^{ix}- e^{-ix})/(2i)$ and cos(x) as $(e^{ix}+ e^{-ix})/2$ so everything is in terms of the exponential?

I tried that, but couldn't figure out where to go with it.

 

[Avodyne]

There is no algebraic solution to this.

 

[gulfcoastfella]

There is no algebraic solution to this.

As I suspected... is there a reason or proof or reference that you can link me to?

Thanks.