Question

Solve for x

sin(x-20) = cos(x)

Answer 1

`x=(6480^(\circ \:)n+1980^(\circ \:))/(36)`

`x=(36\pi n+11\pi )/(36)`

Explanation:

We can use the identity:
`\cos \left(x\right)=\sin \left(90^(\circ \:)-x\right)` to simplify this into:
`\sin \left(x-20^(\circ \:)\right)=\sin \left(90^(\circ \:)-x\right)`

The identity sin(x) = sin(y) -->
`x=y+2\pi n,\:x=\pi -y+2\pi n`, can be used to simplify this further:

`x-20^(\circ \:)=90^(\circ \:)-x+360^(\circ \:)n` (Add 20 to both sides)

`x=-x+360^(\circ \:)n+110^(\circ \:)` (Add x to both sides)

`2x=360^(\circ \:)n+110^(\circ \:)` (Divide both sides by 2)

`(2x)/(2)=(360^(\circ \:)n)/(2)+(110^(\circ \:))/(2)` (Simplify)

`x=(6480^(\circ \:)n+1980^(\circ \:))/(36)` If required, we can convert this into radians:

`x=(36\pi n+11\pi )/(36)`