Linear Function with Trig!

Let $f(x)=px+b$ . If $f(\sin(x))=\cos(x)$ and $f(\cos(x))=\sin(x)$ , then $\sin(x)-\cos(x)=p(\cos(x)-\sin(x))+(b-b)$ which means that $p=-1$ and $f(x)=-x+b$ . Next, $\tan(x)=-\tan(x)+b$ so $\tan(x)=0.5b$ or $\sin(x)=0.5b*\cos(x)$ . If $\sin(x)=m$ and $\cos(x)=\sqrt{1-m^2}$ , then a system of equations can be formed.

After substitution, the equation $2m^4+4m^3+m^2-4m+1=0$ can be derived. Using the Intermediate Value theorem and a Scientific Calculator, we get that $0.3<m<0.4$ and $0.5<m<0.6$ . This means that $0.3<\sin(x)<0.4$ and $0.5<\sin(x)<0.6$ . We can check that the former case does not work by substitution. Taking the arcsine of both sides of $0.5<\sin(x)<0.6$ , we get that $3<0.1x<3.6869$ meaning that $3<0.1x<4$ . This means $a=3$ and $b=4$ so $ab=12$ which is the final answer.