If 9, x, 718 are in arithmetic progression, and 9, y, 518 are also in arithmetic progressi…

Question

If $	an\dfrac{\pi}{9}, x, 	an\dfrac{7\pi}{18}$ are in arithmetic progression, and $	an\dfrac{\pi}{9}, y, 	an\dfrac{5\pi}{18}$ are also in arithmetic progression, then $|x-2y|$ is equal to:

Accepted Answer

$$x = {1 \over 2}\left( {\tan {\pi  \over 9} + \tan {{7\pi } \over {18}}} \right)$$

and $$2y = \tan {\pi  \over 9} + \tan {{5\pi } \over {18}}$$

If we interpret the angles in degrees (as suggested by the numbers 20, 50, and 70), we have :

$$x = \frac{1}{2} \left( \tan 20^\circ + \tan 70^\circ \right),$$

and

$$2y = \tan 20^\circ + \tan 50^\circ.$$

The expression for $|x - 2y|$ is then :

$$|x - 2y| = \left|\frac{\tan 20^\circ + \tan 70^\circ}{2} - \left( \tan 20^\circ + \tan 50^\circ \right)\right|.$$

$$|x - 2y| = \left|\frac{\tan 20^\circ + \tan 70^\circ - 2 \tan 20^\circ - 2 \tan 50^\circ}{2}\right|,$$

which simplifies to :

$$|x - 2y| = \left|\frac{\tan 70^\circ - \tan 20^\circ - 2 \tan 50^\circ}{2}\right|.$$

We know,

$$
\begin{aligned}
& \tan 70=\frac{\tan 20+\tan 50}{1-\tan 20 \tan 50} \\
&\Rightarrow  \tan 70-\tan 70 \cdot \tan 20 \tan 50=\tan 20+\tan 50 \\
&\Rightarrow \tan 70-\tan 50-\tan 20-\tan 50=0 \\
&\Rightarrow \tan 70-\tan 20-2 \tan 50=0
\end{aligned}
$$

$$ \therefore $$ $$|x - 2y| = \left|\frac{\tan 70^\circ - \tan 20^\circ - 2 \tan 50^\circ}{2}\right|$$ = $$\left| {{0 \over 2}} \right|$$ = 0

Solutions · 1