The intensity I I\,I of a laser beam passing through a specific optical filter is modeled by the equation I=5cos2α+sin2αI = 5\cos^2 \alpha + \sin 2\alphaI=5cos2α+sin2α, where α \alpha\,α is the angle of incidence.
Given that I=2I = 2I=2, show that
2tan2α−2tanα−3=0 2\tan^2 \alpha - 2\tan \alpha - 3 = 0 2tan2α−2tanα−3=0Hence, find all possible values of α \alpha\,α in the range 0<α<2π 0 < \alpha < 2\pi\,0<α<2π for which the intensity is 2 units. Give your answers to two decimal places.
Determine the values of x x\,x in the interval 0<x<π3\displaystyle 0 < x < \frac{\pi}{3}0<x<3π such that
5cos2(3x+π4)+sin(6x+π2)=2 5\cos^2 \left( 3x + \frac{\pi}{4} \right) + \sin \left( 6x + \frac{\pi}{2} \right) = 2 5cos2(3x+4π)+sin(6x+2π)=2Give your answers to one decimal place.
