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

Question

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.

Answer 1

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 $|2k+1|\leq 7$...

Answer 2

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?