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

Click For Summary

Discussion Overview

The discussion revolves around using initial value problems (IVPs) to prove the identity cos²(x) + sin²(x) = 1. Participants explore various methods and approaches, including differential equations and matrix forms, while expressing uncertainty about the best path to the solution.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant suggests starting with the differential equations du/dt = -v and dr/dt = u, with initial conditions u(0) = 1 and v(0) = 0, but expresses uncertainty about how to proceed.
  • Another participant mentions Euler's formula and suggests that a Taylor expansion might be relevant, recalling a similar problem from a previous course.
  • A different participant indicates familiarity with Euler's method but lacks understanding of deriving Euler's formula, questioning the relevance of Taylor expansion in this context.
  • One participant proposes using integration by parts and suggests differentiating sin²(x) + cos²(x) to show it is a constant function, but acknowledges uncertainty about the IVP requirement.
  • Another participant reiterates the need to use IVPs specifically, expressing frustration at the constraints of the assignment.
  • A later reply introduces a matrix form of the system and discusses diagonalizing the matrix by finding eigenvalues and eigenvectors, while also emphasizing the importance of showing that u² + v² is constant.

Areas of Agreement / Disagreement

Participants generally agree on the need to use IVPs to prove the identity, but multiple competing views and methods are presented, and the discussion remains unresolved regarding the best approach.

Contextual Notes

Participants express uncertainty about the relevance of certain mathematical techniques, such as Taylor expansion and integration by parts, and there are unresolved questions about the steps needed to solve the differential equations in the context of the IVP.

Who May Find This Useful

Students and educators interested in the application of initial value problems in proving trigonometric identities, as well as those exploring the connections between differential equations and trigonometric functions.

Shadowl08
Messages
3
Reaction score
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.
 
Physics news on Phys.org
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.
 
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.
 
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.
 
yea sadly instead of integration by parts it specifies IVP-Initial value problem. Thanks for trying though :)
 
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.
 
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]
 

Similar threads

MHB -b.2.2.26 IVP min value y'=2(1+x)(1+y^2); y(0)=0
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Solving Differential Equation Using Reduction of Order
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad A question about variables of integration
  • · Replies 3 ·
Replies
3
Views
7K
Undergrad On vibrating string and differentiating infinite sum
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Verifying properties of Green's function
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Helmholtz Equation in Cartesian Coordinates
  • · Replies 11 ·
Replies
11
Views
4K
MHB Solving an IVP for a system of ODEs
  • · Replies 10 ·
Replies
10
Views
4K
Undergrad Characteristic curves for ##u_t + (1-2u)u_x = -1/4, u(x,0) = f(x)##
  • · Replies 2 ·
Replies
2
Views
4K
Undergrad Determining linear indepedence/dependence of a set of functions
  • · Replies 11 ·
Replies
11
Views
4K
MHB How to Solve the Differential Equation x'=x+sin(t)?
  • · Replies 3 ·
Replies
3
Views
2K