Find All Solutions Sinx 1 2 in the Domain
Solve sinx > -1/2 or find the domain of f(x)=1/(sqrt(1+2sinx))
Answer
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Solution :
For f(x) to get defined, we must have 1 + 2 sin `x gt 0` or `sin x gt -(1)/(2)` <br> As the function sin x has fundamental period `2pi`, it is sufficient to solve the inequality first on the interval of length `2pi`, and then get the solution set by adding numbers of the form `2npi, n in Z`, to each of the solutions obtained on that interval. <br> Thus, let us solve this inequality on the interval `[-(pi)/(2), (3pi)/(2)]`. <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/CEN_GRA_C03_S01_042_S01.png" width="80%"> <br> From the above figure, `sin x gt - (1)/(2)` when `- (pi)/(6) lt x lt (7pi)/(6)` <br> Thus, on generalizing, the above solution becomes `2npi - (pi)/(6) lt x lt 2npi + (7pi)/(6) , n in Z`
Transcript
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| 00:00 - 00:59 | hello one in this question we have call finax greater than minus 1 by 2 or find the domain of FX equal to 1 by under root 1 + 2 sin x right so first of all this question same here is not accept of War because if you find domain there now let me effect light effect has won by under root 1 + 2 sin x not finding its normal I can say my denominator part cannot be equal to zero right so I can say 1 + 2 sin x cannot be equal to zero it means I next cannot be equal to minus 1 by 2 now due to square root I can say 1 + 2 sin x must be greater than zero so here we get sin x later than minus 1 by 2 which is the same thing here right that's why I am saying that there is no difference between voting for hair now let's all this first of all we have to make our diagram so |
| 01:00 - 01:59 | that we can get a clear understanding of it now this is a very good question this is zero this is my fibre to this is my this is my 3 pi by 2 and this is my 25 famous for their also minus 5 by 2 minus 5 and we have -35 2 and minus 25 know the value is 1 and -1 here no we know that sin x belongs to sin x belongs to -1 to 1 now at fibre to we have minus 1 minus 5 by 28 at minus 5 by 2 the value of sin x is what - 1 and 8 + 5 by 2 we have so if I make a diagram hear of the finax then my diagram will be look like this 15 2001 352 |
| 02:00 - 02:59 | it is -4 and on 25 it is against you same as for here also right something like this graph of y = 2 sin x what we need we need sin x greater than we need flynax greater than minus 1 by 2 is minus 1 by 2 let's suppose this is my - 1 by 2 log this value comes at minus 5 by 6 write this value comes at minus 5 by 6 and again we have minus one by two here as at + 5 by 6 + 1 by 2 let's suppose this is my 5 by 6 a.m. so this is one by two positive One by Two now we have what we have 5 by 6 5 + 5 + |
| 03:00 - 03:59 | 5 by 6 37648 15 by 60 again but a value this value is 75 essay this point we have 75/6 now the period of sin x is what head of sin x is 25 right and it lies between minus 5 by 2 + 5 by 2 x minus 5 by 2 + 5 by 2 will complete its one cycle right so we can say that we need this value which is greater than minus 1 by to let this value we as per question we required that values Vishnu going to complete your one cycle right so the shaded portion is your one cycle this value is greater than minus 1 by 2 the value of sin x is greater than minus 1 by 2 is going on the yellow portion so I can say from the graph that sin x |
| 04:00 - 04:59 | belongs to minus 5 by open bracket because I cannot take finax as minus one by two 275 price is right write this is the value of x not fine XL this is the value of x 2 X belongs to minus 5 by 6 to 7 5 by 6 and I can write it as X belongs to 75 virus my message also in general form in general form I can write it has to end Pi minus 5 by 6 2 X 22 and Pi plus 75/6 where for all and belongs to interior for all and belongs to integer and this is my final answer this is the domain so this is my final answer thank you |
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