Graph of $7\cos x+5\sin x=2k+1$

The question is to find the number of integral values of k for which the equation $$7\cos x+5\sin x=2k+1$$ has a solution.

I have solved it by converting the LHS into $$\sqrt{74}\sin (x+\alpha)$$, where $$\alpha= \arctan(7/5)$$. And then making an inequality with RHS.

But I wonder how to solve this question graphically.

• Please use MathJax. – saulspatz Jun 22 at 13:32

You can write $$a\sin x+b \cos x$$ in a form $$A\sin (x+\phi)$$ where $$A=\sqrt{a^2+b^2}$$ In your case $$A= \sqrt{74}<9$$, so $$|2k+1|\leq 7$$...
Hint: Write the left-hand side as $$\sqrt{74}\left(\frac{7}{\sqrt{74}}\cos(x)+\frac{5}{\sqrt{74}}\sin(x)\right)$$ so $$\sin(\phi)=\frac{7}{\sqrt{74}}$$ $$\cos(\phi)=\frac{5}{\sqrt{74}}$$ Can you proceed?