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Nairaland Forum / Nairaland / General / Education / Nairaland Mathematics Clinic (492783 Views)
Mathematics Clinic / 2015 Cowbell Mathematics Champion, Akinkuowo Honoured By School. / Lead City University Clinic Welcomes First Ever Baby Since 10 Years Of Opening (2) (3) (4)
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Re: Nairaland Mathematics Clinic by indoorlove(m): 4:29pm On Jan 12, 2013 |
Richiez:thanks once again! Sorry, are u a student of unibuja ? My e-mail:adams_itconsult@123mail.org |
Re: Nairaland Mathematics Clinic by Nobody: 5:18pm On Jan 12, 2013 |
indoorlove: thanks richiez. Pls no 2. Richiez, good job! All the gurus in the house respect! @Indoorlove, richiez did a good job. I may post an alternative method for you later. |
Re: Nairaland Mathematics Clinic by Nobody: 5:57pm On Jan 12, 2013 |
x-fire: I used .gif. 3^x + 2^x = 5^x To plot these graphs, you will need a table of values for 5^x and 3^x + 2^x To create the table, you will give x a range of values, say from -2 to 2 and calculate the corresponding values of each of the equation by substituting each value of x in the equations. For the graph 5^x, Lets subtitute the value of x each from -2 to 2 When x = -2 5^x = 5^(-2) = 1/5^2 = 1/25 3^x + 2^x = 3^(-2) + 2^(-2) =1/9 +1/4 = 13/36 When x = -1 5^x = 5^(-1) = 1/5^1 = 1/5 3^x + 2^x = 3^(-1) + 2^(-1) =1/3 +1/2 = 5/6 When x = 0 5^x = 5^(0) = 1 3^x + 2^x = 3^0 +2^0 = 1+ 1 = 2 When x =1 5^x = 5^(1) = 5 3^x + 2^x = 3^1 + 2^1 = 5 When x = 2 5^x = 5^2 = 25 3^x + 2^x = 3^2 + 2^2 = 13 And so on..... x = -2 , -1, 0, 1, 2 5^x = 1/25, 1/5, 1, 5, 25 3^x + 2^x = 13/36, 5/6, 2, 5, 13 If you plot these two graphs with the above table, they will intersect at x = 1, i.e when their values at x = 1 are the same. So the value of x at their point of intersection is your answer, which is 1. If you notice at x = 1, 3^x + 2^x = 5 and 5^x = 5 meaning that the two graphs will intersect at that point. To the best of my knowledge, that's the only way that equation can be solved! 2 Likes |
Re: Nairaland Mathematics Clinic by Nobody: 6:08pm On Jan 12, 2013 |
doubleDx: Ok thanks sir, I get it now. |
Re: Nairaland Mathematics Clinic by Richiez(m): 6:09pm On Jan 12, 2013 |
doubleDx: simple and straight forward, kudos |
Re: Nairaland Mathematics Clinic by Richiez(m): 6:12pm On Jan 12, 2013 |
indoorlove: thanks once again! Sorry, are u a student of unibuja ? My e-mail:adams_itconsult@123mail.org yep i'm a student of Uniabuja, i got your email addy and i'll reply soonest |
Re: Nairaland Mathematics Clinic by Nobody: 6:19pm On Jan 12, 2013 |
Thanks for opening this thread @richiez. 1. Prove that cosec^2(x)cos(4x) - 15sin^2(x) + 6 = (cosec x + 3sin x)(cosec x - 3sin x) 2. Evaluate ∫ cos^2(x) sin^3(x) dx Sorry, I missed some numbers in the first question. Corrected. Thanks |
Re: Nairaland Mathematics Clinic by indoorlove(m): 7:30pm On Jan 12, 2013 |
Richiez:thanks bruv, i'm a unibuja student too.....u and others are doing a great job here! |
Re: Nairaland Mathematics Clinic by indoorlove(m): 7:32pm On Jan 12, 2013 |
doubleDx:ok, thanks. |
Re: Nairaland Mathematics Clinic by Richiez(m): 7:52pm On Jan 12, 2013 |
Ghettoguru: Thanks for opening this thread @richiez. let me help with no 2 for now; ∫cos^2(x)sin^3(x)dx can be rewritten as; ∫cos^2(x)[sin(x)sin^2(x)dx........................(1) from trig identities, sin^2(x) = 1 - cos^2(x)..............(2) we now put eqn(2) in eqn(1) ∫cos^2(x)[1-cos^2(x)]sin(x)dx....................(3) let us make u=cosx..............................(4) this implies du/dx = -sinx and dx= -du/sinx................................(5) putting eqn(4) and (5) in eqn(3) we get ∫cos^2(x)[1-cos^2(x)]sin(x)dx = ∫u^2[1-u^2]sin(x)*-du/sinx = -∫u^2[1-u^2]du = ∫[u^4-u^2]du = (u^5)/5 - (u^3)/3 + C But recall that u=cosx therefore the proper solution is; 1/5[cos^5(x)] - 1/3[cos^3(x)] + C 1 Like |
Re: Nairaland Mathematics Clinic by indoorlove(m): 8:10pm On Jan 12, 2013 |
Ghettoguru: Thanks for opening this thread @richiez.QUESTION 2: Int cos^2(x)sin^3(x)dx. The above could be written as Int[(cos^2(x) sin^2(x)}sin(x)]dx. Now let u=cos(x), du=-sin(x)dx, and dx=-du/sin(x). Sin^2(x)=1-cos^2(x) -trig. Therefore: -Int[u^2{1-u^2}sin(x).du/sin(x)= -Int[u^2-u^4]du -[u^3/3-u^5/5]. Rem: u=cos(x), therefore, the final answer is -[cos^3(x)/3-cos^5(x)]+c ANS |
Re: Nairaland Mathematics Clinic by indoorlove(m): 8:11pm On Jan 12, 2013 |
Ghettoguru: Thanks for opening this thread @richiez.QUESTION 2: Int cos^2(x)sin^3(x)dx. The above could be written as Int[(cos^2(x) sin^2(x)}sin(x)]dx. Now let u=cos(x), du=-sin(x)dx, and dx=-du/sin(x). Sin^2(x)=1-cos^2(x) -trig. Therefore: -Int[u^2{1-u^2}sin(x).du/sin(x)= -Int[u^2-u^4]du -[u^3/3-u^5/5]. Rem: u=cos(x), therefore, the final answer is -[cos^3(x)/3-cos^5(x)]+c ANS. Note:Int means integral sign! 1 Like |
Re: Nairaland Mathematics Clinic by Nobody: 8:20pm On Jan 12, 2013 |
Thanks @richiez and indoorlove 1. Prove that cosec^2(x)cos(4x) - 15sin^2(x) + 6 = (cosec x + 3sin x)(cosec x - 3sin x) There were some errors in question 1. I was coping two different questions at the same time ![]() ![]() |
Re: Nairaland Mathematics Clinic by Nobody: 9:23pm On Jan 12, 2013 |
Richiez: Kool, nice work bruv! |
Re: Nairaland Mathematics Clinic by Nobody: 9:31pm On Jan 12, 2013 |
Ghettoguru: Thanks @richiez and indoorlove @ math wan tear my head lmao Here is your solution=> Cosec^2 x cos 4x - 15sin^2 x + 6 = (cosec x + 3sin x)(cosec x - 3sin x) To solve this, you need the know of some basic trig identities like cos^2 x + sin^2 x = 1, Cos(A + B ) = cos A cos B - sin A sin B, cosec x = 1/sin x etc From LHS Cosec^2 x cos 4x - 15sin^2 x + 6, lets look for an expression for cos 4x in terms of sin x. We are looking towards sinx and cosec x, so it's best we simplify in terms of sin x and also cosec x is the inverse of sin x. cos 4x = cos (2x + 2x) Applying Cos(A + B ) = cos A cos B - sin A sin B Cos (2x+2x) = cos 2x cos 2x - sin 2x sin 2x ......(1) Since cos 2x = cos (x + x) = cos^2 x - sin^2 x = 1 - sin^2 x - sin^2 x cos 2x = 1 - 2sin^2 x And sin 2x = sin x cos x + cos x sin x = 2sin x cos x Substituting cos 2x and sin 2x back in equation (1) cos 4 x => = (1 - 2sin^2 x)(1 - 2sin^2 x) - 2sin x cos x (2sin x cos x) = 1 - 4sin^2 x + 4sin^4 x - 2sin^2 x cos^2 x substitute cos^2 x = 1 - sin^2 x = 1 - 4sin^2 x + 4sin^4 x - 2sin^2 x ( 1 - sin^2 x) = 1 - 4sin^2 x + 4sin^4 x - 2sin^2 x + 2sin^4 x = 1 - 6sin^2 x + 6sin^4 x Thus Cosec^2 x cos 4x - 15sin^2 x + 6 => (1 - 6sin^2 x + 6sin^4 x)(1/sin^2 x) - 15sin^2 x + 6 = (1/sin^2 x) - 6 + 6sin^2 x - 15sin^2 x + 6 = cosec^2 x = cosec^2x - (3^2)sin^2 x Applying the difference of two squares (a^2 - b^2) = (a - b)(a + b) Thus Cosec^2 x cos 4x - 15sin^2 x + 6 => = (cosec x + cos x)(cosec x - cos x) LHS = RHS QED |
Re: Nairaland Mathematics Clinic by Richiez(m): 9:43pm On Jan 12, 2013 |
Bravo! |
Re: Nairaland Mathematics Clinic by biolabee(m): 11:22pm On Jan 12, 2013 |
bravo! |
Re: Nairaland Mathematics Clinic by Nobody: 7:03am On Jan 13, 2013 |
Who can differentiate btw d kernel of homomorphism of a group and that of a ring |
Re: Nairaland Mathematics Clinic by Nobody: 9:17am On Jan 13, 2013 |
Thanks y'all @ Richiez, doubleD and indoorlove. Appreciated! |
Re: Nairaland Mathematics Clinic by Bumpad: 1:23pm On Jan 13, 2013 |
pls can someone help me with the following question ....find the periods of the folliwing question sinx/x , /sinx/,/sinxcosx/,cos(wx+alpha) |
Re: Nairaland Mathematics Clinic by Nobody: 8:54pm On Jan 13, 2013 |
Prove that there are no simple groups of order 63. |
Re: Nairaland Mathematics Clinic by Fetus(m): 10:42pm On Jan 13, 2013 |
Can sumone solve dis 4 me....IF Tan(x+y)=4/3 and tanx=1/2, evaluate tany....kindly show d workings.....another is...Simplify sinB +sin3B +sin5B/ cosB + cos3B + cos5B...note B=beta |
Re: Nairaland Mathematics Clinic by Richiez(m): 9:18am On Jan 14, 2013 |
Fetus: Can sumone solve dis 4 me....IF Tan(x+y)=4/3 and tanx=1/2, evaluate tany....kindly show d workings.....another is...Simplify sinB +sin3B +sin5B/ cosB + cos3B + cos5B...note B=beta solution coming up shortly |
Re: Nairaland Mathematics Clinic by Nobody: 9:21am On Jan 14, 2013 |
Fetus: Can sumone solve dis 4 me....IF Tan(x+y)=4/3 and tanx=1/2, evaluate tany....kindly show d workings.....another is...Simplify sinB +sin3B +sin5B/ cosB + cos3B + cos5B...note B=beta To solve this question, apply the trig identity for tan (x + y) which is => (tan x + tan y)/(1 - tan x tan y) Since tan (x + y) = 4/3 (tan x + tan y)/(1 - tan x tan y) = 4/3 cross multiplying yields => 4 (1 - tan x tan y) = 3 (tan x + tan y) 4 - 4tan x tan y = 3tan x + 3tan y Collecting like terms => 3tan y + 4tan x tan y = 4 - 3tan x tan y (3 + 4tan x) = 4 - 3tan x tan y = (4 - 3tan x)/(3 + 4tan x) Now remember that tan x = 1/2, substituting yields => tan y = [4 - 3(1/2)]/[3 + 4(1/2)] tan y = [4 - 3/2]/5 tan y = (5/2) .(1/5) :. tan y = 1/2. I'm on phone, I will post the solution to question 2 later. |
Re: Nairaland Mathematics Clinic by Richiez(m): 9:24am On Jan 14, 2013 |
Fetus: Can sumone solve dis 4 me....IF Tan(x+y)=4/3 and tanx=1/2, evaluate tany....kindly show d workings.....another is...Simplify sinB +sin3B +sin5B/ cosB + cos3B + cos5B...note B=beta but for the 2nd part of the question, is it (sinB+sin3B+sin5B)/(cosB+cos3B+cos5B) or sinB+sin3B+(sin5B/cosB)+Cos3B+cos5B pls reply quickly for prompt solution |
Re: Nairaland Mathematics Clinic by Richiez(m): 9:28am On Jan 14, 2013 |
doubleDx: oh there you are...good job doubledx |
Re: Nairaland Mathematics Clinic by Fetus(m): 10:40am On Jan 14, 2013 |
Richiez:(sinB+sin3B+sin5B)/(cosB+cos3B+cos5B)...yea dis one |
Re: Nairaland Mathematics Clinic by Fetus(m): 10:49am On Jan 14, 2013 |
doubleDx:....Tank u so much...God bless u... |
Re: Nairaland Mathematics Clinic by kasbeats(m): 11:28am On Jan 14, 2013 |
tanks guys .......i really do appreciate ur efforts...buh i hope someone can explain to me d Laplace transformation........wats it about.....and is dere any awoite on dis thread,buzz me on 07030195065,add me on 2go-kareem806 |
Re: Nairaland Mathematics Clinic by Nobody: 8:45pm On Jan 14, 2013 |
Show that the ring of polynomials F[x ] is an Euclidean domain , if Ø(f) = deg F |
Re: Nairaland Mathematics Clinic by Nobody: 9:17pm On Jan 14, 2013 |
Fetus: (sinB+sin3B+sin5B)/(cosB+cos3B+cos5B)...yea dis one (sinB+sin3B+sin5B )/( cosB+cos3B+cos5B ) To simplify this, you need a know some basic trig identities and expressions like sin2B = 2sinBcosB, cos2B = cos2B - sin2B or 1 - 2sin2B or 2cos2B - 1, cos2B + sin2B = 1 Lets simplify the numerator => sinB + sin3B + sin5B sin 3B = sin (2B + B ) = sin 2B cos B + cos 2B sin B Subsituting the expressions for sin2B and cos2B Sin3B = cosB ( 2sinBcosB ) + sin B (cos2B - sin2B ) = sin B (1- 2sin2 B ) + 2sin B (1 - sin2 B ) sin 3B = 3sin B - 4 sin3 B cos3B = cos (2B + B ) = cos 2B cos B - sin2B sin B = cos B(1 - 4sin2 B ) Now lets simplify sin 5B => Sin (2B + 3B ) = sin2B cos3B + cos 2Bsin3B Subsituting the expressions for sin2B, sin3B, cos2B and cos3B Sin5B = 2sinBcosB.cosB(1 - 4sin2 B ) + (1 - 2sin2B )( 3sinB - 4sin3B ) Expanding and collecting like terms => Sin 5B = 16sin5B - 20sin3B + 5sinB Now, the numerator sinB + sin3B + sin5B => = sinB + 3sin B - 4 sin3B + 16sin5B - 20sin3B + 5sinB Numerator = 16sin5B - 24sin3B + 9sinB Lets take the denominator cosB+ cos3B + cos5B => We already know cos3B = cos B ( 1 - 4sin3B ) if we simplify it in terms of cos B, cos 3B = 4cos3B - 3cos B Cos5B = cos (2B + 3B ) = cos2B cos3B - sin2B sin 3B Subsituting the expressions for sin2B, sin3B, cos2B and cos3B cos5B = (2cos2B -1 )(4cos3B - 3cosB ) - 2sinBcosB (3sinB - 4sin3B ) Expanding and collecting like terms, Cos5B = 16cos5B - 20cos3B + 5cosB :. The denominator cosB+ cos3B + cos5B => = cosB + 4cos3B - 3cosB + 16cos5B - 20cos3B + 5cosB Collecting like terms=> Denominator = 16cos5B -16cos3B + 3cosB Therefore the fraction (sinB+sin3B+sin5B )/( cosB+cos3B+cos5B ) => (16sin5B - 24sin3B + 9sinB )/ (16cos5B -16cos3B + 3cosB ) = sinB/cosB [ (16sin4B - 24sin2B +9)/(6cos4B -16cos2B + 3) ] = tanB [ (16sin4B - 24sin2B +9)/(6cos4B -16cos2B + 3) ] The above expression could still be simplified further either in terms of sinB only or cos B only. But I'll stop there. I hope that helps. |
Re: Nairaland Mathematics Clinic by TOSINACCA(m): 10:10pm On Jan 14, 2013 |
Please kindly solve this jamb question for me: b=a +cp and r=ab + 1/2cp^2, express b^2 in terms of a, c and r |
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