The value of sin-n-sin3-n-sin5 -n- to n terms is

We know that, sin A+sin (A+D)+⋯+sin (A+(n 1)D) nbsp;= sinnD2sinD2[sin(2A+(n 1)D2)] Now, using the above fornula sinπn+sin3πn+sin5 πn+⋯ to n terms nbsp; ...

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Standard XIII

Mathematics

Trigonometric Ratios of Multiples of an Angle

The value of ...

Question

The value of $sin \frac{π}{n} + sin \frac{3 π}{n} + sin \frac{5 π}{n} + \dots to n terms$ is

0

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1

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\sqrt{3}

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2

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Solution

The correct option is A $0$
We know that,
$sin A + sin (A + D) + \dots + sin (A + (n - 1) D) = \frac{sin \frac{n D}{2}}{sin \frac{D}{2}} [sin (\frac{2 A + (n - 1) D}{2})]$
Now, using the above fornula
$sin \frac{π}{n} + sin \frac{3 π}{n} + sin \frac{5 π}{n} + \dots to n terms = \frac{sin \frac{n \cdot 2 π}{2 \cdot n}}{sin \frac{2 π}{2 \cdot n}} ⎡ ⎢ ⎢ ⎢ ⎣ sin ⎛ ⎜ ⎜ ⎜ ⎝ \frac{2 \cdot \frac{π}{n} + (n - 1) \cdot \frac{2 π}{n}}{2} ⎞ ⎟ ⎟ ⎟ ⎠ ⎤ ⎥ ⎥ ⎥ ⎦ = 0$

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The value of sinπn+sin3πn+sin5πn+⋯ to n terms is

The correct option is A 0We know that, sinA+sin(A+D)+⋯+sin(A+(n−1)D)=sinnD2sinD2[sin(2A+(n−1)D2)] Now, using the above fornula sinπn+sin3πn+sin5πn+⋯ to n terms=sinn⋅2π2⋅nsin2π2⋅n⎡⎢ ⎢ ⎢⎣sin⎛⎜ ⎜ ⎜⎝2⋅πn+(n−1)⋅2πn2⎞⎟ ⎟ ⎟⎠⎤⎥ ⎥ ⎥⎦=0

The value of $sin \frac{π}{n} + sin \frac{3 π}{n} + sin \frac{5 π}{n} + \dots to n terms$ is