Basic algebra?

1. Apr 23, 2008

Swest

Hello all

Can anybody see a way to analytically find x in the expression:

cos(x) = exp(-2x)

By inspection x=0 is obvious, and numerically we find x=1.5232 is also a solution, but is there a way to find these values by rearranging the above expression? It's one of those that looks simple but isn't, any ideas?

Many thanks

2. Apr 23, 2008

HallsofIvy

Staff Emeritus
That is not "simple" algebra. What you are attempting to do is solve an equation involving two different transcendental functions. In general there is no simple way to do that.

3. Apr 23, 2008

Swest

Thanks, I suspected as much. Always nice to be reassured that one isn't missing something obvious.

4. Apr 23, 2008

mathwonk

graphing them is instructive as it shows the graphs meet infinitely often, at points that become closer and closer to the zeroes of cos(x),

i.e. the solutions approach closer and closer to the integral multiples of pi/2.

5. Apr 23, 2008

ice109

that make sense seeing as for large x the exponential is vanishing.

6. Apr 24, 2008

Swest

Yes, I did a quick matlab plot of the two and saw the trend that you describe, which again can be surmised from inspection, indeed the 2nd solution is very close to pi/2.