If x=(5)/(12) for some x ( ,(3 )/(2)), then 7x( (13x)/(2)+ (13x)/(2))+ 7x( (13x)/(2)- (13x…

Question

If $\cot x=\frac{5}{12}$ for some $x\in (\pi ,\frac{3\pi }{2})$, then $\sin 7x(\cos \frac{13x}{2}+\sin \frac{13x}{2})+\cos 7x(\cos \frac{13x}{2}-\sin \frac{13x}{2})$ is equal to

Accepted Answer

$\cot x=\frac{5}{12},x\in (\pi ,\frac{3\pi }{2})$ means 3rd quadrant.

To find the value of:

$\sin 7x(\cos \frac{13x}{2}+\sin \frac{13x}{2})+\cos 7x(\cos \frac{13x}{2}-\sin \frac{13x}{2})$

Simplify:

$=\sin 7x\cos \frac{13x}{2}+\sin 7x\sin \frac{13x}{2}+\cos 7x\cos \frac{13x}{2}-\sin \frac{13x}{2}\cos 7x$

Rearranging terms and use $\sin A\cos B-\cos A\sin B=\sin (A-B)$ and $\cos A\cos B+\sin A\sin B=\cos (A-B)$ :

$\begin{gathered} =\sin 7x\cos \frac{13x}{2}-\sin \frac{13x}{2}\cos 7x+\cos 7x\cos \frac{13x}{2}+\sin 7x\sin \frac{13x}{2} \ =\sin (7x-\frac{13x}{2})+\cos (7x-\frac{13x}{2}) \ =\sin \frac{x}{2}+\cos \frac{x}{2} \end{gathered}$

Since $x\in (\pi ,\frac{3\pi }{2})$, then $\frac{x}{2}\in (\frac{\pi }{2},\frac{3\pi }{4})$.

Let $L=\sin \frac{x}{2}+\cos \frac{x}{2}$.

In the second quadrant, $\frac{x}{2}$ makes $L$ positive because $\sin \frac{x}{2}>|\cos \frac{x}{2}|$.

Squaring:

${L}^{2}={\sin }^{2}\frac{x}{2}+{\cos }^{2}\frac{x}{2}+2\sin \frac{x}{2}\cos \frac{x}{2}$

${L}^{2}=1+\sin x=1+\frac{1}{\csc x}=1+\frac{1}{-\sqrt{1+{\cot }^{2}x}}$, Since $x$ is in the 3rd quadrant, $\sin x$ must be negative

${L}^{2}=1+\frac{1}{-\sqrt{1+\frac{25}{144}}}(\cot x=\frac{5}{12})$

${L}^{2}=1-\frac{1}{\sqrt{\frac{169}{144}}}=1-\frac{12}{13}=\frac{1}{13}$

$L=\frac{1}{\sqrt{13}}$

Solutions · 1