The value of sin-7-sin2- 7-sin3- 7 is equal to -

The correct option is C 12cotπ14 sinπ/7+sin2π/7+sin3π/7 2sin3π/14cos( π/4)+2sin3π/14cos3π/14 {sin C+sin D= 2 sinC+D/2cosC D/2 sin 2x= ...

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Standard XII

Mathematics

Definition of Functions

the value of ...

Question

the value of $s i n \frac{π}{7} + s i n \frac{2 π}{7} + s i n \frac{3 π}{7}$ is equal to :-

c o t \frac{π}{14}

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\frac{1}{2} c o t \frac{π}{14}

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t a n \frac{π}{14}

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\frac{1}{2} t a n \frac{π}{14}

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Solution

The correct option is C $\frac{1}{2} c o t \frac{π}{14}$
$s i n \frac{π}{7} + s i n \frac{2 π}{7} + s i n \frac{3 π}{7}$
$2 s i n \frac{3 π}{14} c o s (- \frac{π}{4}) + 2 s i n \frac{3 π}{14} c o s \frac{3 π}{14}$
${s i n C + s i n D = 2 s i n \frac{C + D}{2} c o s \frac{C - D}{2}$
$s i n 2 x = 2 s i n x c o s x}$
${c o s (- θ) = c o s θ}$
$= 2 s i n \frac{3 π}{14} (c o s \frac{π}{14} + c o s \frac{3 π}{14})$
$= 2 s i n \frac{3 π}{14} [2 c o s \frac{π}{7} c o s \frac{π}{14}]$
$= 4 c o s (\frac{π}{2} - \frac{3 π}{14}) c o s \frac{2 π}{14} c o s \frac{π}{14}$ ${s i n θ = c o s (\frac{π}{2} - θ)}$
$= 4 c o s \frac{4 π}{14} . c o s \frac{2 π}{14} . c o s \frac{π}{14} \times \frac{s i n \frac{π}{14}}{s i n \frac{π}{14}}$
$= \frac{2 c o s \frac{4 π}{14} . c o s \frac{2 π}{14} . s i n \frac{2 π}{14}}{s i n \frac{π}{14}}$ ${s i n 2 θ = 2 s i n θ c o s θ}$
$= \frac{c o s \frac{4 π}{14} . s i n \frac{4 π}{14}}{s i n \frac{π}{14}} \times \frac{2}{2}$
$= \frac{s i n \frac{8 π}{14}}{2 s i n \frac{π}{14}} = \frac{c o s (\frac{π}{2} - \frac{8 π}{14})}{2 s i n \frac{π}{14}} = \frac{c o s \frac{π}{14}}{2 s i n \frac{π}{14}} = \frac{1}{2} c o t \frac{π}{14}$
Option (B) is correct.

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Similar questions

Q. The Value of

sin \frac{π}{14} \cdot sin \frac{3 π}{14} \cdot sin \frac{5 π}{14}

$c o s \frac{π}{7}$ . $c o s \frac{3 π}{7}$ . $c o s \frac{5 π}{7}$ are the roots of the equation $8 x^{3} - 4 x^{2} - 4 x + 1 = 0$ . Then the value of $s i n \frac{π}{14} . s i n \frac{3 π}{14} . s i n \frac{5 π}{14}$ is

Q. The value of

sin \frac{π}{14} \cdot sin \frac{3 π}{14} \cdot sin \frac{5 π}{14} \cdot sin \frac{7 π}{14} \cdot sin \frac{9 π}{14} \cdot sin \frac{11 π}{14} \cdot sin \frac{13 π}{14}

is equal to

Q. Find the value of

sin \frac{π}{14} . sin \frac{3 π}{14} . sin \frac{5 π}{14} . sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} . sin \frac{13 π}{14}

Q. Solve :

{sin}^{2} \frac{π}{7} . {sin}^{2} \frac{5 π}{14} . {sin}^{2} \frac{8 π}{7} . {sin}^{2} \frac{9 π}{14}

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the value of sinπ7+sin2π7+sin3π7 is equal to :-

the value of $s i n \frac{π}{7} + s i n \frac{2 π}{7} + s i n \frac{3 π}{7}$ is equal to :-