How do I find out the value of $p$ for $\sin p + \cos p = 0$? I tried doing this by putting $$\frac{ \sin p}{\sqrt2} +\frac{ \cos p}{\sqrt2} = 0 $$ which implies $$ \sin(p + π/4) = \sin 0 $$ which implies $p+ π/4 = nπ $. Now according to the question I'm solving $p = 2πt/T$. I need to get the relation $t = 3T/8$. How do I get this? Am I going the right way?

$$\sin x + \cos x = 0$$
$$\Rightarrow \sin^2 x + 2 \sin x \cos x + \cos^2 x = 0$$
$$\Rightarrow \sin(2x) + 1 = 0$$
$$\Rightarrow \sin(2x) = -1$$
Since we know that $\sin x = -1$ when $x = \frac{3\pi}{2} + 2\pi n$, then $\sin(2x) = -1$ when $x = \frac{3\pi}{4} + \pi n $.
