Some challenge questions before Monday :-) (1 Viewer)

[ToRnaDo]

Some challenge questions heheh...

1.solve sin ^nx + cos^nx = 2 ^(2-n)/2

2. solve 3arctg(x) - arctg (3x) = pi/2

3. pi^/sinx^(0.5)/ = /cosx/


If u can solve them???? Well i will difinitely call u guys teacher

 

[Affinity]

1.)

n=0,2 all x are solutions

n=1 only x=pi/4

Pi/4 + (m/2)Pi for even n
Pi/4 + 2mPi for odd n
and there's another set of solutions for odd n

 

[Constip8edSkunk]

argh crap.... im only gonna attmp the 2nd 1...

3arctanx-arctan3x=pi/2
arctanx-arctan3x=pi/2 - 2arctanx
tan[arctanx-arctan3x] = tan [pi/2 - 2arctanx]
[x-3x]/[1+3x^2]= cot[2arctanx]

u=2arctanx x=tan(u/2)
cot u = [1-x^2]/[2x]

[x-3x]/[1+3x^2] = [1-x^2]/[2x]
[edit:...mistake]
-[2x][2x] = [1+3x^2][1-x^2]
-4x^2 = 1+2x^2-3x^4
3x^4 - 6x^2 - 1 = 0

solve and check back in2 the oiginal eqn...

 

[Affinity]

2.) sqrt(1 + sqrt(4/3))

 

[Affinity]

the quartic should be 3x^4 -6x^2 -1 ..
you mucked up the trig somewhere

3arctan(x) = pi/2 + arctan(3x)

tan(LHS) = tan(RHS)

(3x - x^3)/1-3x^2 = -1/3x

etc

 

[Affinity]

last one

pi > 1
so LHS >= 1
but RHS <= 1

so only solutions are RHS = LHS = 1

sin(sqrt(x)) = 0, /cos(x)/ = 1

sqrt(x) = npi, x=mpi

n^2 pi^2 = mpi
n^2 pi = m
pi is transcedental

so only solution will be n=m = 0

so x=0 is only solution

 

[Constip8edSkunk]

ah yeah... knew i made a mistake some where...

 

[Affinity]

I didn't finish with (1)

if n is even the it could be established by the power mean inequalities that LHS >= RHS qith equality iff sin(x) =cos(x)

gets nastier with odd powers though, coz there are solution in the second and fourth quad

 

[blah]

wtf is arc

 

[Affinity]

inverse... textbooks write tan^-1

 

[fitz33]

Originally posted by ToRnaDo
Some challenge questions heheh...

you forgot this one:
P=NP

it's a shoe-in for question eight

 

[tonberry_kun]

i look at these and i think... "im dead"

 

[ricenoodles]

Originally posted by tonberry_kun
i look at these and i think... "im dead"

here here

 

[MyLuv]

ToRnaDo can U give us the solution for the 1st one ?
All I can do was finding the graph and the interval for other sets of solution ?