Question

Find the general solution of the equation $\sin \pi x+\cos \pi x=0$. Also find the sum of all solutions in $[0,100]$.

Answer 1

$\sin \pi x=-\cos \pi x$

$\tan \pi x=-1$, let $\pi x=y$

$\Rightarrow y\in [0,100\pi ]$

As $\tan x$ repeats itself after $x\in [0,\pi ]$

and in $y\in [0,\pi ]$, $\tan y=-1$ for $y=\frac{3\pi }{4}$, hence solutions would be $y=\frac{3\pi }{4},\frac{7\pi }{4},\frac{11\pi }{4}$…..

$\Rightarrow x=\frac{3}{4},\frac{7}{4},\frac{11}{4},...\dots ..,\frac{399}{4}$ [As $4x$ would $g\theta \le 400$]

[And one term before every $100$ would appear]

Sum of A.P$=\frac{n}{2}(a+l)=\frac{n}{2}(\frac{3}{4}+\frac{399}{4})$

for $n,\{a_n=a+(n-1\left)d\right\}$

$\Rightarrow \frac{399}{4}=\frac{3}{4}+(n-1)$

$\Rightarrow \frac{396}{4}=n-1$

$\Rightarrow 99+1=n$

$\Rightarrow n=100$

$\Rightarrow$ Sum$=50\left(\frac{402}{4}\right)=5025$

And general solution, $x=\frac{(4n-1)}{4}$, $n=1,2,$….