Answer to Question #273 in Trigonometry for Nick

Question

Find the general form of the solutions of the equation x such that sin x = cos 5x

Example: for the equation tanx=1, the solutions can be described by x = pi/4 + n*pi, n is an integer.

Answer 1

sin x = cos5x
sin x = cos(pi/2 - x)
cos(pi/2 - x) = cos(5x)
cos(pi/2 - x) - cos(5x) = 0
-2sin(( pi/2-x+5x)/2 ) sin ((pi/2-x-5x)/2)=0
-2sin (pi/4+2x) sin (pi/4 -3x)= 0
х=pi*n/2-pi/8 or х=pi/12-pi*n/3, nEZ