A physical experiment measures the spectral intensity III of a pulsed laser relative to its frequency shift sss. The measurements are modeled by the parametric equations
s=5+4sint s = 5 + 4 \sin t s=5+4sint I=187+cos2t I = \frac{18}{7 + \cos 2t} I=7+cos2t18where −π2≤t≤π2-\frac{\pi}{2} \le t \le \frac{\pi}{2}−2π≤t≤2π.
Show that the relationship between intensity and frequency shift can be expressed as
I=144(13−s)(s+3)p≤s≤q I = \frac{144}{(13 - s)(s + 3)} \quad p \le s \le q I=(13−s)(s+3)144p≤s≤qwhere ppp and qqq are constants to be found.
Hence, determine a Cartesian equation for this relationship in the form
I=as+b+cs+dp≤s≤q I = \frac{a}{s + b} + \frac{c}{s + d} \quad p \le s \le q I=s+ba+s+dcp≤s≤qwhere a,b,ca, b, ca,b,c and ddd are constants.
