Question

(a) Convert `cos2x-sqrt3sin2x` as a cosine ratio only; (b) solve the equation `y=f(x)=cos2x-sqrt3sin2x-1` and (c) draw the graph of the function to check result?

Answer 1

Please see below.

Explanation:

(a) `cos2x-sqrt3sin2x=2(cos2x xx1/2-sin2x xxsqrt3/2)`

= `2(cos2xcos(pi/3)-sin2xsin(pi/3))`

= `2cos(2x+pi/3)`

(b) `y=f(x)=cos2x-sqrt3sin2x-1`, will cut `x`-axis where `y=0` i.e. where `cos2x-sqrt3sin2x=1` or `2cos(2x+pi/3)=1` i.e. when

`cos(2x+pi/3)=1/2=cos(2npi+-pi/3)` i.e.

`x+pi/6=npi+-pi/6`, where `n` is an integer or `{-pi,-pi/3,0,(2pi)/3,pi}` in `[-pi,pi]` (choose `n=-1,0,1` for this).

It will cut `y`-axis where `x=0` i.e. `y=0` - note that it means that `f(x)` passes through origin.

(c) The graph of `y=f(x)=cos2x-sqrt3sin2x-1` appears as follows:

graph{cos(2x+pi/3)-1/2 [-5.05, 4.95, -3.09, 1.91]}