aDdditional MaThs

[usman]

solve da trigonometric equation

sin (2X) = cos(X)
plz show da steps also

 

[PlanetMaster]

Piece of cake :mrgreen:

Substitute Sin(2x) = 2Sin(x)Cos(x)
therefore 2Sin(x)Cos(x) = Cos(x)
and 2Sin(x)Cos(x) - Cos(x) = 0
Taking Cos(x) common: Cos(x)[2Sin(x) - 1] = 0

Cos(x) = 0
so x = π/2 or x = 3π/2

[2Sin(x) - 1] = 0
2Sin(x) = 1
Sin(x) = 1/2
so x = π/6 or x = 5π/6

 

[Talha]

@ PlanetMaster: How sin(2x) became 2sin(x)cos(x) ??

 

[PlanetMaster]

Its a trigonometrical identity to remember. Luckily its derivation is not in your syllabus.

 

[usman]

thankx a lot
now can u plz do this derivation (piece of cake) 4 me also (as a cHallenGe)
plz differentiate the expression in terms of x.
d/dx (x^x^x)
NB: x^x^x stands for x to the power of x to the power of x.

:mrgreen:
also plz solve dis logarithmic equation 4 me (without using graphs) and find da exact value of x in terms of log

2^x + x = 50

NB: 2^x stands for 2 raised to the power x

 

[PlanetMaster]

Derivation Challenge:
Since Sin(A+B) = Sin(A)Cos(B) + Cos(A)Sin(B),
Sin(2x) = Sin(x+x) = Sin(x)Cos(x) + Cos(x)Sin(x) = 2Sin(x)Cos(x)


Differentiation question:
Suppose that y = x^g(x) and you want to differentiate y with respect to x.
First take the natural log of both sides and use the properties of the log functions to get

ln = ln( x^g(x)) = g(x) ln(x)

Now differentiate both sides

y'/y = g'(x) ln(x) + g(x)/x

and hence

y' = y [ g'(x) ln(x) + g(x)/x ] that is
y' = [x^g(x)] [ g'(x) ln(x) + g(x)/x ]

In your problem g(x) = x^x so you are going to have to use this method again to find g'(x)


Logarithmic equation:
I'm pretty sure it can't be done using logarithms.
You can either do it by graph, Iteration or Newton raphson.
The answer for x is 5.476.

 

[usman]

thanx a lot 1ce again . m not very gud at chemistry can u plz balance dis chemical equation 4 me
Cu + HNO3 = Cu(NO3)2 + H2O + NO

http://www.youtube.com/watch?v=N5kkwVoAtkc

How to solve tan x dy/dx + y = sec x?
HELP

d/dx (x^x^x) = x^x^x [x^(x-1) + x^x ( lnx + 1 ) lnx ]
and 2^x + x = 50
x = 5.476495256242

by da way (sorry 2 say) the exact value of x can be found using
x = 50 - ProductLog[1125899906842624 Log[2]]/Log[2]

 

[PlanetMaster]

Chemical equation balance:
3Cu + 8HNO3 = 3Cu(NO3)2 + 4H2O + 2NO

Solve Tan(x) [dy/dx] + y = Sec(x)?
Solve what?
I think it requires us to make [dy/dx] the subject and then integrate it to get a equation of y & x.

Derivative of x^x^x:
If its the final derivative, it would be exactly the same from my method in my previous post.

Solve 2^x + x = 50 using logarithm:
Can you please show steps?