Finding a functional connection.

In summary, the given equations are u=arcsin(x)+arccos(y) and v=xsqrt(1-y^2)+ysqrt(1-x^2). The functional connection between u and v is v= sin(u) or u= arcsin(v). This can be shown by using the derivatives of u and v and simplifying the resulting expression. Alternatively, it can also be derived by using the trigonometric identity sin(a+b)= sin(a)cos(b)+cos(a)sin(b).
  • #1
MathematicalPhysicist
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im given:
u=arcsin(x)+arccos(y)
v=xsqrt(1-y^2)+ysqrt(1-x^2)
and i need to find the functional connection between u and v.

i know that:
v'_x/u'_x=dv/du
and i have got:
dv/du=sqrt(1-x^2)sqrt(1-y^2)-xy
now i need to show the rhs as a function of v or u, obviously of v should be much easier, the problem is i don't know how to simplify it.

i mean i tried adding and substracting terms, but with no success can someone help me on this?

thanks.
 
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  • #2
loop quantum gravity said:
im given:
u=arcsin(x)+arcsin(y)
v=xsqrt(1-y^2)+ysqrt(1-x^2)
and i need to find the functional connection between u and v.

i know that:
v'_x/u'_x=dv/du
and i have got:
dv/du=sqrt(1-x^2)sqrt(1-y^2)-xy
now i need to show the rhs as a function of v or u, obviously of v should be much easier, the problem is i don't know how to simplify it.

i mean i tried adding and substracting terms, but with no success can someone help me on this?

thanks.
i fixed my mistake, the first equation is u=arcsin(x)+arcsin(y)
can now someone help me on this?
thanks.
 
  • #3
How about

[tex] v=x\left(1-y^{2}\right)\frac{\partial u}{\partial y} +y\left(1-x^{2}\right)\frac{\partial u}{\partial x} [/tex]

Is this what you were looking for?

Daniel.
 
  • #4
i need to find a functional connection like u=u(v) or v=v(u), or in simple terms v is a function of u or otherwise.
 
  • #5
[itex]cos(arcsin(x))= \sqrt{1- sin^2(arcsin(x)}= \sqrt{1- x^2}[/itex]

Since sin(a+ b)= sin(a)cos(b)+ cos(a)cos(b), sin(arcsin(x)+ arcsin(y))= sin(arcsin(x))cos(arcsin(y))+ cos(arcsin(x))sin(arcsin(y)).

That is, sin(arcsin(x)+ arcsin(y))= [itex]x\sqrt{1- y^2}+ y\sqrt{1-x^2}[/itex].

In other words, v= sin(u)!
 
  • #6
yes i also found it by using derivatives we know that:
v'_x=sqrt(1-y^2)-xy/sqrt(1-x^2)
and u'_x=1/sqrt(1-x^2)
so dv/du=sqrt(1-x^2)sqrt(1-y^2)-xy
(sqrt(1-x^2)sqrt(1-y^2)-xy)^2=1-v^2
 

Related to Finding a functional connection.

1. What is a functional connection?

A functional connection is a relationship between two or more entities that allows them to interact and influence each other in a meaningful way. It can refer to biological connections between different parts of an organism, physical connections between objects, or even abstract connections between ideas or concepts.

2. How do you find a functional connection?

Finding a functional connection involves conducting research and analysis to identify the relationship between two or more entities. This can be done through experiments, observations, or data analysis. It may also require collaboration with other experts in the field to gain a deeper understanding of the connection.

3. Why is it important to find functional connections?

Functional connections are important because they help us understand the world around us and how different elements interact with each other. By identifying functional connections, we can gain insights into complex systems and make predictions about how they may behave in the future. This can be useful in many fields, including biology, engineering, and social sciences.

4. What are some techniques for finding functional connections?

There are various techniques for finding functional connections, depending on the specific field of study. Some common methods include correlation analysis, network analysis, and statistical modeling. In biology, techniques such as gene expression analysis and brain imaging can also be used to identify functional connections.

5. Can functional connections change over time?

Yes, functional connections can change over time. This can be due to various factors, such as environmental changes, evolution, or learning. In some cases, functional connections may also be intentionally modified through interventions or treatments. It is important to continuously study and monitor functional connections to understand how they may change and adapt in different contexts.

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