High School Complicated equation and Simple equation for the Same Curve?

[pairofstrings]

I can always make an equation more complicated. For instance y=x can be rewritten as y=(x+1)(sin^2x+cos^2x) -1.

Does y = x and y=(x+1)(sin^2x+cos^2x) -1 have the same meaning or is it the same graph but only the expressions are changing?

 

[Mark44]

Does y = x and y=(x+1)(sin^2x+cos^2x) -1 have the same meaning

Yes. The two equations are equivalent -- for a given value of x, both equations produce the same y value.

or is it the same graph but only the expressions are changing?

Yes to that, also. The graphs would be exactly the same. The only difference is that the expression on the right in the 2nd equation is unsimplified.

 

[pairofstrings]

The graphs would be exactly the same. The only difference is that the expression on the right in the 2nd equation is unsimplified.

I didn't think that I have online graphing calculators for producing graphs for these equations. Sorry about that.

Okay, so the unsimplified 2nd equation is destined to become 'x' on simplification? Like this: y = x? Is it possible to get anything else other than 'x' on simplification from this unsimplified 2nd equation?

 

[hutchphd]

Yes. Making something "simpler" is more difficult: you reach the point where you cannot continue but you can never be certain. You can complexify ad infinitum

 

[pairofstrings]

The two equations are equivalent -- for a given value of x, both equations produce the same y value.

the unsimplified 2nd equation is destined to become 'x' on simplification? Like this: y = x? Is it possible to get anything else other than 'x' on simplification from this unsimplified 2nd equation?

Does y = x and y=(x+1)(sin^2x+cos^2x) -1 have the same context?
y = x may have context not similar to y = ( x + 1)( sin^2x + cos^2x ) - 1...

First one is linear equation (straight lines) and the second equation is trigonometric equation (side lengths and angles).

What could y = x be?
What could y=(x+1)(sin^2x+cos^2x) -1 be?

What does it mean if y is x and y is (x+1)(sin^2x+cos^2x) -1?

Thanks!

 

[Mark44]

Does y = x and y=(x+1)(sin^2x+cos^2x) -1 have the same context?
y = x may have context not similar to y = ( x + 1)( sin^2x + cos^2x ) - 1...

I don't know what "same context" means in regard to equations. The two equations are equivalent, meaning that they both represent exactly the same points, and their graphs are exactly the same.

First one is linear equation (straight lines) and the second equation is trigonometric equation (side lengths and angles).

No, they are both linear equations, meaning that the graph of each equation is a straight line. Both graphs pass through the origin, and both graphs have a slope of 1.

What could y = x be?
What could y=(x+1)(sin^2x+cos^2x) -1 be?

Already answered -- both are straight lines through the origin with a slope of 1.

 

[pairofstrings]

I don't know what "same context" means in regard to equations. The two equations are equivalent, meaning that they both represent exactly the same points, and their graphs are exactly the same.

Context here is that there is sine, cosine in y = (x + 1)(sin²x + cos²x) -1 but there is nothing like that in y = x...

 

[Mark44]

Context here is that there is sine, cosine in y = (x + 1)(sin²x + cos²x) -1 but there is nothing like that in y = x...

Right, but "context" is pretty meaningless when you're talking about equations. A specific graph can have any number of equations that represent it, in part due to whether the expressions making up the equation have been fully simplified or not.

So, what reason will make me use sin²x + cos²x in the equation?

Normally you wouldn't. The person who gave this as an example was just making a point that the equation y = x can be written many ways.

In yet another way, the equation $y = \frac 1 2 \ln(e^{2x})$ is equivalent to y = x, if x > 0. IOW, the graph of this equation aligns perfectly with the graph of y = x in the first quadrant, and excluding the origin.

 

[Frabjous]

I might have missed the suggestion, but things could get ugly with an infinite series expansion in terms of your favorite orthogonal functions.

 

[pairofstrings]

The person who gave this as an example was just making a point that the equation y = x can be written many ways.

y = (x + 1)(sin²x + cos²x) -1 is an equation which appears as y = x when simplified.

Identity:
sin²x + cos²x = 1

What was the scenario for which Trigonometric functions like sine, cosine in equation y = (x + 1)(sin²x + cos²x) -1 were used?

I ask this question because y = x and y = (x + 1)(sin²x + cos²x) -1 both have similar graph, but the terms in the expressions are different. The terms are not same but the graph is, that is why please let me know the difference between y = x and y = (x + 1)(sin²x + cos²x) -1.

 

[Frabjous]

y = (x + 1)(sin²x + cos²x) -1 is an equation which appears as y = x when simplified.

Identity:
sin²x + cos²x = 1

What was the scenario for which Trigonometric functions like sine, cosine in equation y = (x + 1)(sin²x + cos²x) -1 were used?

I ask this question because y = x and y = (x + 1)(sin²x + cos²x) -1 both have similar graph, but the terms in the expressions are different. The terms are not same but the graph is, that is why please let me know the difference between y = x and y = (x + 1)(sin²x + cos²x) -1.

I do not understand your question.
sin²x + cos²x =1 so the equation simplifies to (x+1)-1=x

 

[Mark44]

What was the scenario for which Trigonometric functions like sine, cosine in equation y = (x + 1)(sin²x + cos²x) -1 were used?

As I already said, someone earlier in the thread wanted to make the point that expressions can be written in many different ways, but have the same value.

I ask this question because y = x and y = (x + 1)(sin2x + cos2x) -1 both have similar graph, but the terms in the expressions are different.

The graphs are not just similar -- they are exactly the same.

The terms are not same but the graph is, that is why please let me know the difference between y = x and y = (x + 1)(sin²x + cos²x) -1.

Again, there is no real difference - the first one is just the simplified form of the latter one.

 

[pairofstrings]

As I already said, someone earlier in the thread wanted to make the point that expressions can be written in many different ways, but have the same value.

But what is this equation talking about?: y = (x + 1)(sin²x + cos²x) -1?

What is the phenomenon?

I agree that y = (x + 1)(sin²x + cos²x) -1 is y = x on simplification, and someone earlier in the thread wanted to make the point that expressions can be written in many different ways, but have the same value but what is sin²x, cos²x doing in the equation?

 

[Frabjous]

But what is this equation talking about?: y = (x + 1)(sin²x + cos²x) -1?

What is the phenomenon?

What is the equation y=x talking about? What is its phenomena?

The answers to your questions are the same as the answers to mine.

My answers are nothing and nothing. An equation is a mathematical abstraction that we map reality to.

 

[Mark44]

We seem to be going around in circles. You said this in post #20:

As FactChecker points out the curve can be as simple as y = 1/x but what I am trying to know is if this curve can have a complicated equation; something more than just y = 1/x and I would get the same curve.

Yes it can have a more complicated equation. If I multiply the right side of the equation by any expression whose value is always 1 (such as $\sin^2(x) + \cos^2(x)$) or add some expression whose value is always 0, I will get a new equation that is equivalent to the simpler one I started with -- exactly the same graph.

But what is this equation talking about?: y = (x + 1)(sin²x + cos²x) -1?

Who cares what it is "talking about"? All that matters is that this equation is equivalent to the much simpler equation y = x. To get this equation, we can add 0 in the form of 1 + (-1) to x to get x + 1 - 1, then multiply the x + 1 part by 1 in the form of $\sin^2(x) + \cos^2(x)$.

What is the phenomenon?

It doesn't have to represent any phenomenon. Someone can write down an equation that has no physical significance whatsoever, but so what? You are being overly concerned about something that really isn't worth all of that angst.

I agree that y = (x + 1)(sin²x + cos²x) -1 is y = x on simplification, and someone earlier in the thread wanted to make the point that expressions can be written in many different ways, but have the same value but what is sin²x, cos²x doing in the equation?

It is multiplying a part of the right hand side by 1, which is always legal to do.

 

[pairofstrings]

I have equation of heart curve: x⁶ + 3x⁴y - 3x⁴ + 3x²y⁴ - x²y³ - 6x²y² + 3x² + y⁶ - 3y⁴ + 3y² = 1.

1. Heart curve with trigonometric functions:
x = 16 sin³
y = 13cost - 5cos(2t) - 2cos(3t) - cos(4t).

2. Psy curve with trigonometric functions.

Last thing I want to know is can I build any object with trigonometric functions as well?

Thanks.

 

[Frabjous]

I have equation of heart curve: x⁶ + 3x⁴y - 3x⁴ + 3x²y⁴ - x²y³ - 6x²y² + 3x² + y⁶ - 3y⁴ + 3y² = 1.

Heart curve with trigonometric function:
x = 16 sin³
y = 13cost - 5cos(2t) - 2cos(3t) - cos(4t).

Last thing I want to know is can I build any object with trigonometric functions as well?

Thanks.

Yes to within reason. It is called a Fourier series.

 

[pairofstrings]

Yes to within reason.

What is "Yes to within reason"? Does it mean that I can build only few types of objects with Fourier Series, not all objects?

 

[Frabjous]

What is "Yes to within reason"?

For example, a discontinuous function will have ringing near the discontinuity or a multi-valued function. I think that you could get around these by defining regions of applicability by defining regions of applicability for multiple series.

 

[Mark44]

I have equation of heart curve: x⁶ + 3x⁴y - 3x⁴ + 3x²y⁴ - x²y³ - 6x²y² + 3x² + y⁶ - 3y⁴ + 3y² = 1.

1. Heart curve with trigonometric functions:
x = 16 sin³
y = 13cost - 5cos(2t) - 2cos(3t) - cos(4t).

This is a different question from the one you posted in this thread, and is related to another thread you started a month or so ago. Your question of this thread apparently has been answered, so I'm closing this thread.

In your first equation above, you have an equation involving x and y. In the equations involving trig functions, x and y are given as parametric equations. A given curve can have different equations that generate the points on the curve, depending on whether the equations have been simplified or not (discussed in this thread) or depending on whether the curve is define in terms of parametric equations (heart equation with trig functions). A given curve can also have different sets of equations depending on the coordinates being used, such as if it's defined Cartesian coordinates or polar coordinates.

For example, a circle of radius 1/2 with center at (1/2, 0) can be defined by the Cartesian equation $(x - 1/2)^2 + y^2 = 1$ or by the polar equation $r = \cos(\theta)$.