if sin (x+10) = cos (4x-5), what is x

Question

determe what x is

Answer 1

Assuming degrees, sin(x+10)=sin(90-(4x-5)), x+10=90-4x+5, 5x=85, x=17°. There are other solutions. For example x=89° because sin(x+10)=sin(360+90-(4x-5))=sin(450-4x+5), so x+10=455-4x, 5x=445, x=89°.

Also, sin(x+10)=sin(90+4x-5), x+10=90+4x-5, -75=3x, x=-25 which is the same as 360-25=335°.

A more general answer is x=17+72n and x=95+120n where n is any integer.

This gives us the following angles (between 0 and 360°) for x:

17, 89, 95, 161, 215, 233, 305, 335 degrees.