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If sin 2x+cosx=0 , then which among the ...

If sin 2x+cosx=0 , then which among the following is /are true ?
I. cosx =0 , II. `sinx=-(1)/(2)` , III `x=(2n+1)(pi)/(2),ninZ` , IV. `x=npi+(-1)^(n)(7pi)/(6),ninZ`

A

I is true

B

I and II are true

C

I , II and II are true

D

All are true

Text Solution

Verified by Experts

The correct Answer is:
D
`sin2x+cosx=0`
`rArr2sinxcosx+cosx=0 " "[becausesin2x=2sinc cosx]`
`rArrcosx(2sinx+1)=0`
`rArrcosx=0orsinx=- (1)/(2)`
when cosx =0,
Then`x=(2n+1)(pi)/(2)`
When `sinx=-(1)/(2)`,
Then , `sinx=-"sin"(pi)/(6)`
`sinx=sin(pi+(pi)/(6)) " " [becausesin(pi+theta)=-sintheta]`
`sinx=sin:"(7pi)/(6)`
`rArrx=npi+(-1)^(n)(7pi)/(6),ninZ` ...
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