How to derive pythagorean identity?

  • Thread starter Thread starter colormyworld
  • Start date Start date
  • Tags Tags
    Derive Identity
AI Thread Summary
To derive the Pythagorean identity sin²(x) + cos²(x) = 1, one approach involves using the angle addition formula for cosine, specifically setting y such that x + y = 0, which simplifies to cos(x + y) = 1. Understanding that cosine is an even function and sine is an odd function is crucial in this derivation. A more intuitive explanation connects the identity to the unit circle, where the coordinates correspond to cos(x) and sin(x), illustrating the relationship to Pythagorean theorem. Some participants express confusion about the choice of y over x and the relevance of even and odd functions in the derivation. Overall, visualizing the identity through the unit circle can clarify its geometric significance.
colormyworld
Messages
2
Reaction score
0
I got some precalc review to prepare for calc, and after hours of doing the packet, I'm on the last problem set...but it's all about derivatives which we never touched on last year.

Homework Statement


I'm supposed to derive sin^2 + cos^2 = 1


Homework Equations


It says to use cos 0 =1, cos (x+y) = cos x cos y - sin x sin y, but I have no idea how to use these.


please help! I'm absolutely clueless at math :frown:
 
Physics news on Phys.org
All you need to do is choose y so that x+y = 0 and thus cos (x+y) = 1 and plug your values into the right hand side of the equation you're given. You will also need to know about cos being an even function and sin being an odd function.
 
Thanks! I got it
 
Its not the best derivation of the result because it leaves you wondering what Pythagoras has to do with it. Its best to derive this result from the unit circle where the x coordinate is given by cos(x) and the y coordinate by sin (x) then it becomes immediately apparent where Pythagoras comes in.
 
That interseting because i just derived pythagoras c^2=a^2+b^2 formula from an ellipse, very enlightening.
 
Last edited:
sorry but can anyone please explain it to me again? i don't understand why you have to choose a value for y and not x and i don't see how even/odd functions will be used
 

Thread 'Finding the number of ways to arrange identical balls in a circle (3 different colors)'

I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
View full post »

Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
View full post »

Similar threads

How to Solve a Trigonometry Equation Using Identities and Alternative Methods?
Replies
7
Views
2K
Does the Pythagorean Identity Hold for sin^2(3x) + cos^2(3x)?
Replies
4
Views
2K
Deriving the Matrix for a 3 dimensional rotation
Replies
9
Views
3K
How Do You Simplify Trigonometric Expressions Using Basic Identities?
Replies
5
Views
2K
Quick Trigonometric Identity Question
Replies
5
Views
2K
Trigonometry - Double Angle Identities
Replies
18
Views
2K
Trigonometric Equation Solving: Cosine and Sine Identities for Homework
Replies
5
Views
2K
Trig identities (I think?) for precalc review
Replies
5
Views
2K
Finding a domain for a function
Replies
7
Views
2K
Prove the identity matrix is unique
2
Replies
69
Views
8K

Hot Threads

Recent Insights

Back
Top