The positive integer value of n-3 satisfying the equation1-sin -n -1-sin 2-n -1-sin 3-n is

We have, 1/sinπ/n 1/sin3π/n=1/sin2π/n ⇒sin3π/n sinπ/n/sinπ/nsin3π/n=1/sin2π/n⇒2cos2π/nsinπ/n/sinπ/nsin3π/n=1/sin2π/n ⇒ 2sin2π/ncos2π/n=sin3π/n⇒ sin4π/n sin3π/n= ...

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Byju's Answer

Standard XIII

Mathematics

General Solution of sin theta = sin alpha

The positive ...

Question

The positive integer value of n>3 satisfying the equation
$\frac{1}{s i n (\frac{π}{n})} = \frac{1}{s i n (\frac{2 π}{n})} + \frac{1}{s i n (\frac{3 π}{n})}$ is ___

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Solution

We have, $\frac{1}{s i n \frac{π}{n}} - \frac{1}{s i n \frac{3 π}{n}} = \frac{1}{s i n \frac{2 π}{n}} \Rightarrow \frac{s i n \frac{3 π}{n} - s i n \frac{π}{n}}{s i n \frac{π}{n} s i n \frac{3 π}{n}} = \frac{1}{s i n \frac{2 π}{n}} \Rightarrow \frac{2 c o s \frac{2 π}{n} s i n \frac{π}{n}}{s i n \frac{π}{n} s i n \frac{3 π}{n}} = \frac{1}{s i n \frac{2 π}{n}} \Rightarrow 2 s i n \frac{2 π}{n} c o s \frac{2 π}{n} = s i n \frac{3 π}{n} \Rightarrow s i n \frac{4 π}{n} - s i n \frac{3 π}{n} = 0 \Rightarrow 2 c o s \frac{7 π}{n} s i n \frac{π}{n} = 0 \Rightarrow c o s \frac{7 π}{n} = 0 o r s i n \frac{π}{2 n} = 0 \Rightarrow \frac{7 π}{2 n} = (2 k + 1) s i n \frac{π}{2}) o r \frac{π}{2 n} = 2 k π w h e r e k ϵ Z \Rightarrow n = \frac{7}{2 k + 1} o r n = \frac{1}{4 k}$
( $n = \frac{1}{4 k}$ not possible for any integral value of k)
As n>3, for k=0, we get n=7

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The positive integer value of n>3 satisfying the equation 1sin(πn)=1sin(2πn)+1sin(3πn) is ___

The positive integer value of n>3 satisfying the equation
$\frac{1}{s i n (\frac{π}{n})} = \frac{1}{s i n (\frac{2 π}{n})} + \frac{1}{s i n (\frac{3 π}{n})}$ is ___