Prove cos^2(x)+sin^2(x)=1 using IVP's

In summary, the problem requires using IVPs to prove that cos^2(x) + sin^2(x) = 1. The end result should be du/dt = -v, u(0) = 1 and dr/dt = u, v(0) = 0, but the process to get to this point is unclear. Possible methods include using Euler's formula or identity, Taylor expansion, integration by parts, or solving the differential equation in matrix form by diagonalizing the matrix. The desired quantity to show is u^2 + v^2, and the chain rule can be used to
  • #1
Shadowl08
3
0
Ok so for this problem I have to use IVPs to prove that cos^2(x)+sin^2(x)=1. I know the end result is suppose to be:
du/dt= - v, u(0)=1
dr/dt= u, v(0)=0
but I have no idea how to go about getting to this point.
 
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  • #2
Are you aware of Euler's formula, or identity, or have any idea on how to derive Euler's?

Oh, I think you might have to use a Taylor expansion. I remember doing something similar last quarter, but it isn't quite coming back to me yet.
 
  • #3
I have used Euler's method before but it was based on a program. As for deriving Euler's I am unsure of that. I don't think it would be tayler expansion solely because we learned that after I received an assignment with this question.
 
  • #4
They ask you to do it by integration by parts? What you can do is take the derivative of sin^2x +cos^2x and finding that it is a constant function, then plugging in those values to verify, but as for doing it by IVP I am unsure, I'm sorry.
 
  • #5
yea sadly instead of integration by parts it specifies IVP-Initial value problem. Thanks for trying though :)
 
  • #6
Shadowl08 said:
Ok so for this problem I have to use IVPs to prove that cos^2(x)+sin^2(x)=1. I know the end result is suppose to be:
du/dt= - v, u(0)=1
dr/dt= u, v(0)=0
but I have no idea how to go about getting to this point.
Do you know how to solve a differential equation in matrix form?

Your system can be written as
$$\begin{bmatrix} u \\ r \end{bmatrix}'= \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} u \\ v\end{bmatrix}$$
with the initial condition
$$\begin{bmatrix} u(0) \\ r(0)\end{bmatrix} = \begin{bmatrix} 1 \\ 0\end{bmatrix}$$
Solving this matrix equation involves diagonalizing the matrix by finding its eigenvalues and eigenvectors.
 
  • #7
Shadowl08 said:
Ok so for this problem I have to use IVPs to prove that cos^2(x)+sin^2(x)=1. I know the end result is suppose to be:
du/dt= - v, u(0)=1
dr/dt= u, v(0)=0
but I have no idea how to go about getting to this point.

That's the starting point. The quantity you want to show is constant is [itex]u^2 + v^2[/itex]. So what does the chain rule give you for [tex]
\frac{d}{dt}(u^2 + v^2)[/tex]
 

1. How can I prove the identity cos^2(x) + sin^2(x) = 1 using IVP's?

The proof of this identity using IVP's (Initial Value Problems) is a direct application of the Pythagorean identity for trigonometric functions. By setting up an IVP with initial conditions of cos(x) = 0 and sin(x) = 1, we can show that cos^2(x) + sin^2(x) = 1 is true for all values of x.

2. Why is it necessary to use IVP's to prove this identity?

Using IVP's allows us to prove the identity for all values of x, rather than just a specific value. This is because IVP's involve setting up a general case with initial conditions, which can be applied to any value of x.

3. Can you provide a step-by-step explanation of how to prove cos^2(x) + sin^2(x) = 1 using IVP's?

First, we set up an IVP with initial conditions of cos(x) = 0 and sin(x) = 1. Next, we take the derivative of both sides of the identity, which gives us -sin(x)cos(x) + cos(x)sin(x) = 0. Using the initial conditions, we can see that this equation is true, proving the identity for these specific values of x. To prove it for all values of x, we can use the uniqueness theorem for solutions to differential equations, which states that if two solutions to a differential equation have the same initial conditions, they must be identical.

4. Are there any alternative methods to proving this identity?

Yes, there are other methods such as using the double angle formula for cosine, which states that cos^2(x) = (1 + cos(2x))/2. By substituting this into the identity, we get (1 + cos(2x))/2 + sin^2(x) = 1. From here, we can use algebraic manipulations to show that this equation is true for all values of x.

5. Can this identity be proven without using calculus or differential equations?

Yes, there are other geometric or algebraic proofs for this identity. One method involves using the unit circle and the Pythagorean theorem to show that the sum of the squares of the cosine and sine of an angle is always equal to 1. Another method involves using the exponential form of trigonometric functions and manipulating the equations to show that the identity holds true.

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