Method For Solving Trigonometric Equations by Introducing Auxiliary Argument

Sometimes it is useful to replace expression `acos(x)+bsin(x)` with expression `A cos(x+phi)`, where `A=sqrt(a^2+b^2)`, `sin(phi)=a/(sqrt(a^2+b^2))`, `cos(phi)=b/(sqrt(a^2+b^2))`. In this case `phi` is called auxiliary argument.

Example. Solve equation `8cos(x)+15sin(x)=17`.

Let's divide both sides of equation by `sqrt(8^2+15^2)=17`: `8/17 cos(x)+15/17 sin(x)=1`.

Since `(8/17)^2+(15/17)^2=1`, then there exists such `phi` that `sin(phi)=8/17` and `cos(phi)=15/17`.

So, equation can be rewritten as `sin(phi)cos(x)+cos(phi)sin(x)=1` or `sin(x+phi)=1`.

Therefore, `x+phi=pi/2+2pin,n in Z` or `x=pi/2-phi+2pin,n in Z`.

Since `sin(phi)=8/17` then `phi=arcsin(8/17)` and finally

`x=pi/2-arcsin(8/17)+2pin,n in Z`.