0
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Please use MathJax. $\endgroup$ – saulspatz Jun 22 at 13:32
2
$\begingroup$

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

$\endgroup$
  • $\begingroup$ I already solved like this! $\endgroup$ – Ramit Jun 22 at 14:28
  • $\begingroup$ Ok, so how can I help you? You wanted a comfirmation? Or you want me to delte it or something else? $\endgroup$ – Aqua Jun 22 at 15:00
  • $\begingroup$ I was wondering how to solve it graphically. Like, making two graphs,one of LHS, and one of RHS. And then superimposing them to see the number of solutions. I have solved other questions like this, but couldn't solve this one. $\endgroup$ – Ramit Jun 22 at 15:26
  • $\begingroup$ Why would you do it like that? It is much harder. You are trying to go from Paris to London via New York... $\endgroup$ – Aqua Jun 22 at 15:29
  • $\begingroup$ Ok, I see your point. One last thing. Shouldn't we say 2k+1 is less than or equal to 8, instead of 7? $\endgroup$ – Ramit Jun 22 at 15:41
0
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ I already solved like this! $\endgroup$ – Ramit Jun 22 at 14:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.