Use inverse trig functions to find the smallest two positive solutions of cos(3.5x+5.5) = 0.1263 on [ 0,2π ).

Question

This problem needs two solutions, that are rounded to four decimal places.

Answer 1

cos(3.5x+5.5)=0.1263, 3.5x+5.5=cos^-1(0.1263)=1.444158 radians approx.

Since, for all angles A, cos(A)=cos(-A)=cos(2π-A), cos^-1(0.1263)=2π-1.444158=4.839027.

Therefore, 3.5x+5.5=1.444158+2πn or 4.839027+2πn where n is an integer. 

3.5x=1.444158+2πn-5.5 or 4.839027+2πn-5.5. The smallest positive value of n which allows x to be positive is 1. Therefore, 3.5x=-4.055842+2π=2.227343 or -0.660973+2π=5.622212.

x=2.227343/3.5=0.6364 or 5.622212/3.5=1.6063 approx.

x=0.6364, 1.6063.