Solve for x, 5sin(x) - 3cos(x) = 2 , in the interval 0<x<2pi

5sin(x)-3cos(x)=2

Express in the form Rsin(x-@) = Rsin(x)cos@ - Rcos(x)sin@

Rcos@ =5

Rsin@ = 3    

tan@ = 3/5           @=tan^-1(3/5)= 0.540

R = sqrt(5² +3² )= sqrt(34)

sqrt(34)sin(x-0.540) = 2

sin(x - 0.540) = 0.343

Let Y = x-0.540

sinY = 0.343         in the interval -0.540 < Y < 5.743

Y = sin^-1 (0.343)=0.350

Using CAST diagrams, we obtain 0.350 and pi-0.350 in this interval

Y = 0.350, 2.791

x = 0.350 + 0.540, 2.791 + 0.540

x = 0.890, 3.331