1.

हल कीजिए - (i) ` sin theta +sin 3 theta + sin 5 theta = 0` (ii) `sin 2 theta + sin 4 theta + sin 6 theta = 0`

Answer» (i) दिया है - `sin theta + sin 3 theta + sin 5 theta = 0`
`rArr (sin theta +sin 5 theta) + sin 3 theta = 0`
` rArr 2 sin 3 theta cos 2 theta + sin 3 theta = 0`
`rArr sin 3 theta (2 cos 2 theta +1) = 0`
` rArr sin 3 theta = 0 "या " 2 cos 2 theta + 1 = 0`
यदि `sin 3 theta = 0 rArr 3 theta = n pi " अर्थात" theta = (n pi)/3`
यदि `2 cos 2 theta + 1 = 0`
`rArr cos 2 theta = - 1/2 = - cos . pi/3 = cos (pi - pi/3)`
`2 theta = 2 n pi pm (2pi)/3, n in Z`
`rArr theta = n pi pm pi/3`
इसलिए `theta = (n pi)/3" या " theta = n pi pm pi/3, n in Z`
(ii) दिया है - `sin 2 theta + sin 4 theta + sin 6 theta = 0`
`rArr sin 2 theta+ (sin 4 theta + sin 6 theta) = 0`
`rArr sin 2 theta + 2 sin 5 theta cos theta = 0`
`rArr 2 sin theta cos theta+ 2 sin 5 theta cos theta = 0`
`rArr 2 cos theta(sin 5 theta + sin theta) = 0`
`rArr 2 cos theta(2 sin 3 theta cos 2 theta) = 0`
`rArr cos theta sin 3 theta cos 2 theta = 0`
यदि `cos theta = 0 rArr theta = (2n +1) pi/2`
यदि `sin 3 theta = 0 rArr 3 theta = n pi rArr theta = (npi)/3`
व यदि `cos 2 theta = 0 rArr 2 theta = (2n +1) pi/2 rArr theta = (2n +1) pi/4 , n in Z`


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