Question

Sin (3x+15) = Cos (7x-2) Find value of x

Answer 1

$\bf \textit{Cofunction Identities} \\ \quad \\ \boxed{sin\left(\frac{\pi}{2}-{{ \theta}}\right)=cos({{ \theta}})}\qquad cos\left(\frac{\pi}{2}-{{ \theta}}\right)=sin({{ \theta}}) \\ \quad \\ \quad \\ tan\left(\frac{\pi}{2}-{{ \theta}}\right)=cot({{ \theta}})\qquad cot\left(\frac{\pi}{2}-{{ \theta}}\right)=tan({{ \theta}}) \\ \quad \\ \quad \\ sec\left(\frac{\pi}{2}-{{ \theta}}\right)=csc({{ \theta}})\qquad csc\left(\frac{\pi}{2}-{{ \theta}}\right)=sec({{ \theta}})\\\\ -------------------------------$

$\bf \begin{array}{lcllclll} sin(&90^o-\theta)&=&cos(&\theta)\\ &\uparrow &&&\uparrow \\ &3x+15&&&7x-2\\ &\downarrow \\ &90-(7x-2) \end{array} \\\\\\ 90-(7x-2)=3x+15\implies 90-7x+2=3x+15 \\\\\\ 92-15=10x\implies 77=10x\implies \cfrac{77}{10}=x$