Question

The positive integer of n > 3 satisfying the equation $\frac{1}{\sin \left(\frac{\pi}{\mathrm{n}}\right)}=\frac{1}{\sin \left(\frac{2 \pi}{\mathrm{n}}\right)}+\frac{1}{\sin \left(\frac{3 \pi}{\mathrm{n}}\right)}$ is

Answer 1

Answer:

Solution :

1sin(πn)=1sin(2πn)+1sin(3πn)

⇒1sin(πn)−1sin(3πn)=1sin(2πn)

⇒sin(3πn)−sin(πn)sin(3πn)sin(πn)=1sin(2πn)

⇒2sin(πn)cos(2πn)sin(3πn)sin(πn)⋅sin(2πn)=1

⇒2sin(2πn)cos(2πn)sin(3πn)=1

⇒sin(4πn)sin(3πn)=1

⇒sin(4πn)=sin(3πn)

⇒sin(π−4πn)=sin(3πn)

⇒π−4πn=3πn

=π=7πn

⇒n=7.

Step-by-step explanation:

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Answer 2

Answer:

n=7

Here, is the solution.

In the last step I used the property

sin(pi-x) = sinx