# 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.