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Trigonometry – II — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometry – II2 Marks
MCQ
$\cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}$
  • A
    is equal to zero
  • B
    lies between 0 and 3
  • ✓
    is a negative number
  • D
    lies between 3 and 6
✓

Answer

Correct option: C.
is a negative number
(C)
$\cos \alpha+\cos 2 \alpha+\cos 3 \alpha+\ldots+\cos$ n $\alpha$
$\cos \alpha+\cos (\alpha+\beta)+ \cos (\alpha+2 \beta) +\ldots \ldots+\cos [\alpha+(n-1) \beta]$
$=\frac{\cos \left[\alpha+(n-1) \frac{\beta}{2}\right] \cdot \sin \left(\frac{n \beta}{2}\right)}{\sin \frac{\beta}{2}}$
If $\beta=\alpha$, then
$\cos \alpha+\cos 2 \alpha+\cos 3 \alpha+\ldots \ldots+\cos n \alpha$
$=\frac{\cos \left(\frac{ n +1}{2}\right) \alpha \cdot \sin \left(\frac{ n \alpha}{2}\right)}{\sin \left(\frac{\alpha}{2}\right)}$
Here, $n =3$ and $\alpha=\frac{2 \pi}{7}$
$\therefore \cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7}$
$=\frac{\cos \left(\frac{3+1}{2}\right)\left(\frac{2 \pi}{7}\right) \sin \left(\frac{3 \times 2 \pi}{2 \times 7}\right)}{\sin \left(\frac{2 \pi}{7 \times 2}\right)}$
$=\frac{\cos \left(\frac{4 \pi}{7}\right) \cdot \sin \left(\frac{3 \pi}{7}\right)}{\sin \left(\frac{\pi}{7}\right)}$
Since the values of $\cos \left(\frac{4 \pi}{7}\right), \sin \left(\frac{3 \pi}{7}\right)$ and $\sin \left(\frac{\pi}{7}\right)$ are $- ve ,+ ve$ and $+ve$ respectively.
∴ option $( C )$ is the correct answer.

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