A laser tracking system uses a sensor where the angle xxx (in radians) of the beam relative to a horizontal axis must satisfy the equation:
5sec2x+7tanx=55\sec^2 x + 7\tan x = 55sec2x+7tanx=5
Determine all solutions for x x\,x in the interval −π<x≤0-\pi < x \le 0−π<x≤0, giving your answers to 3 significant figures where appropriate.
In the study of resonance in a harmonic oscillator, an engineer encounters the following expression involving phase angles. Help them prove the identity:
sin7βsin2β−cos7βcos2β≡sin5βsin2βcos2β\frac{\sin 7\beta}{\sin 2\beta} - \frac{\cos 7\beta}{\cos 2\beta} \equiv \frac{\sin 5\beta}{\sin 2\beta \cos 2\beta}sin2βsin7β−cos2βcos7β≡sin2βcos2βsin5β