Differentiating Trig Homework: Find Uncertainty in Measured Angle

[lovemake1]

Homework Statement

we have been doing some error analysis in school, but they were very straight forward for example. centripetal acceleration : Fc = 4pi^2 m R / T^2


however, for my project i must find the uncertainty in the angle that i measured.
the angle is formed by a cardboard sitting on pile of books creating a triangle.
Hypothenus = 33cm
Height = Variable ( Changes according to the stack of the book)

please help ! differentiate theta = inverseSin(Opposite/Hypothenus)

Homework Equations

theta = inverseSin(Opposite/Hypothenus)

The Attempt at a Solution

well since i measurd these distances with a ruler

the uncertainty for hypothenus would be +- 0.005m (last half digit of a number)
and the uncertainty for heght is the same + - 0.005m (last half dight of a number)

how do i use these to differentiate for one another?
which formurla woudl i use pleaes help !

 

[Matterwave]

The derivative of inverse sine can be found in a list for derivatives such as this one: http://en.wikipedia.org/wiki/List_of_differentiation_identities.

 

[lovemake1]

i see that it is 1 ove squareroot of 1 minus x^2.
but at my level of physics and calculus that is just different language to me.
how can x be represented by (Opposite over hypothenus) ??
how will i differentiate them? HELP !

 

[ideasrule]

To do error analysis, you write the whole equation in terms of differentials; you don't differentiate with respect to anything. Here's an example:

if 2x=y^2 and you differentiate both sides, you'd get 2dx=2ydy. Don't write it as dx/dy=y or dy/dx=1/y.

For theta = inverseSin(Opposite/Hypothenus), you should get:

d(theta)=1/sqrt(1-(o/h)^2) * derivative of o/h

What do you get?

 

[lovemake1]

hmm. 1*sqrt(1-(o/h)^-2)
you would get negative exponent.

is this correct so far? please guide me along the way.
i would like to learn this by tom ~ help

 

[ideasrule]

I don't get why it's 1*sqrt(1-(o/h)^-2). The derivative of inverse sine is 1/sqrt(1-x^2), so the first step in deriving theta = inverseSin(Opposite/Hypothenus) is 1/sqrt(1-(o/h)^2).

After that, derive o/h. The answer would be (do*h + o*dh)/h^2, following the quotient rule. do and dh are the errors in the measured lengths and d(theta) is the error in the angle.

 

[lovemake1]

so d(theta) = (do*h + o*dh)/h^2

hmm.. do i sub in the numbers and that is the total uncertainty for theta?
lets say i got 12 degrees from measuring sides with ruler and, after using the above equation i would write

12 degrees + - (do*h + o*dh)/h^2
would this be correct??

but in my pratice sheet which is totally different than trig we have

dF/dPi = 8PimR/T^2 from F = 4Pi^2mR/T^2

and so on with all the other variables...

but with this triangle is this it?
or do i have to differentiate further.. please HELP !