sin-pi-2-4x-x-cos-pi-x-7-x-1-x-

Question Number 188475 by cortano12 last updated on 02/Mar/23

\sin (\frac{π}{2} (4 x + \sqrt{x})) \cos (π (x + 7 \sqrt{x})) = 1

x = ?

Answered by Frix last updated on 02/Mar/23

\sin α \sin β = 1

\Leftrightarrow

\frac{\sin (α - β) + \sin (α + β)}{2} = 1

\Rightarrow

\sin (α - β) = 1 \land \sin (α + β) = 1

\Rightarrow

α - β = \frac{(4 m + 1) π}{2} \land α + β = \frac{(4 n + 1) π}{2} \land m, n \in Z

{\begin{cases} \frac{π}{2} (4 x + \sqrt{x}) - π (x + 7 \sqrt{x}) = \frac{(4 m + 1) π}{2} \\ \frac{π}{2} (4 x + \sqrt{x}) + π (x + 7 \sqrt{x}) = \frac{(4 n + 1) π}{2} \end{cases}

\Leftrightarrow

{\begin{cases} m = \frac{x}{2} - \frac{13 \sqrt{x}}{4} - \frac{1}{4} \\ n = \frac{3 x}{2} + \frac{15 \sqrt{x}}{4} - \frac{1}{4} \end{cases}

\Rightarrow x = h^{2}; h \in N

By trying I got h = 4 k + 1; k \in Z

\Rightarrow

x = {(4 k + 1)}^{2}; k \in Z

Commented by cortano12 last updated on 02/Mar/23

\sin α . \cos β = \frac{\sin (α + β) + \sin (α - β)}{2}

Commented by Frix last updated on 02/Mar/23

Yes, that is what I used .

Facebook Tweet Pin