What is an identity?

[LAVRANOS]

can anybody give me the definition of a trig-identity?
And then the definition of an equation?
Because i think that the relation
\tan^2 x + 1 = \sec^2 x
is not an identity.

[dx]

can anybody give me the definition of a trig-identity?
And then the definition of an equation?
Because i think that the relation
\tan^2 x + 1 = \sec^2 x
is not an identity.


A trig identity is a proposition involving '=' and trigonometric functions that is true for all values of x. An equation is a proposition involving '=' which is not necessarily true for all x. \tan^2 x + 1 = \sec^2 x is an identity because it is true for all x.

[LAVRANOS]

According to your definition when i say: The relation (sinx)^2+(cosx)^2 =1 is beautiful ,this is a trig identity.
Also when i put into my computer ,tan(90),tan270,tan450,tan630 e.t.c and i mean 90 ,270,450,630 deegres i get error which means that the trig (tanx)^2+1=(secx)^2 is not true for all the values of x

[rock.freak667]

According to your definition when i say: The relation (sinx)^2+(cosx)^2 =1 is beautiful ,this is a trig identity.
Also when i put into my computer ,tan(90),tan270,tan450,tan630 e.t.c and i mean 90 ,270,450,630 deegres i get error which means that the trig (tanx)^2+1=(secx)^2 is not true for all the values of x

well in tan(90) is undefined

But it's actually true for all x.

take x=90

tan(90)= \infty and sec(90)=1/cos(90)=\infty

so it is true for x=90 but not very useful.

[dx]

According to your definition when i say: The relation (sinx)^2+(cosx)^2 =1 is beautiful ,this is a trig identity.

Yes, that is a trig identity.


Also when i put into my computer ,tan(90),tan270,tan450,tan630 e.t.c and i mean 90 ,270,450,630 deegres i get error which means that the trig (tanx)^2+1=(secx)^2 is not true for all the values of x

That's because tan is not defined at those values. When I said all values, I meant all values for which the expression is meaningful. For example, x + 0 = x is an identity. But if you put x = !"£%£" on your computer you get an error.

[LAVRANOS]

But you said and i quote:an equation is aproposition involving = which is not necessarily true for all x.So you must decide is it an ID or an equation?
O.K rock.freak667 you mean that (infinity)^2 + 1= (infinity)^2 is true?
When i put into my computer x=90(degrees) and i get an error as an answer it means there is not such x to satisfy the trig equation

[dx]

You get an error because tan x is not defined for x = 90, Just like 5/x is not defined for x = 0. But 5/x = 5/x is still an identity because its true for all values for which it is defined. By all values it is always meant all values for which the function is defined. In my definitions above, "all x" means "all meaningful x". You claim that \tan^2 x + 1 = \sec^2 x is not an identity. By my definition of identity, it is. Whats your definition of identity?

[LAVRANOS]

Allow me to go sleep now ,sorry for the delay but i was looking at darkfire"s 1=i thing .
Tomorrow we will Curry on

[tiny-tim]

For some purposes, infinity is a number.

For trigonometric formulas, I personally regard the range as being R+ (the real numbers plus one point at infinity) rather than R.

It is silly to say that tan90º does not exist when R+ is a perfectly valid number system, and everyone know that tan90º = ∞.

So, for example, the formula tanx = sinx/cosx is an identity, and is valid even when cosx = 0. :smile:

[LAVRANOS]

dx ,i asked you whether the sentance < The relation (sinx)^2 +(cosx)^2 =1 is beautiful > is atrig identity and you said yes .
Do still insist on that?
Also trhere is a great difference BETWEEN the words VALID and TRUE .VALID is an argument ,adeductuion alogical implication while TRUE OR FALSE IS a proposition or a sentance.
SO we can have a VALID argument with FALSE result.
HOW would you define an IDENTITY WITHIN logic(sympolic logic)????

[dx]

dx ,i asked you whether the sentance < The relation (sinx)^2 +(cosx)^2 =1 is beautiful > is atrig identity and you said yes .
Do still insist on that?

No, I said (sinx)^2 +(cosx)^2 =1 is a trig identity. Statements like "I am beautiful" are not propositions unless there's a well defined meaning to "beautiful".


Also trhere is a great difference BETWEEN the words VALID and TRUE .VALID is an argument ,adeductuion alogical implication while TRUE OR FALSE IS a proposition or a sentance.
SO we can have a VALID argument with FALSE result.
HOW would you define an IDENTITY WITHIN logic(sympolic logic)????

I don't know how one would precisely define identity. But everyone knows that things like (tanx)^2 + 1 = (secx)^2 are identities. If you don't agree, tell me what you think an identity is.

[Count Iblis]

See here (http://www.math.upenn.edu/~wilf/AeqB.html) (section 1.5 is devoted to trig identities) :cool:

[CRGreathouse]

Here's a shot at a basic definition of identity.

An identity is an an ordered triple (A, E_1, E_2) where A is an algebraic structure and E_1, E_2 are expressions on A, where expressions are defined recursively as follows:
x is an expression on A
If e_1,e_2,\ldots,e_n are expressions on A, and O is an n-ary operator on A, then O(e_1,e_2,\ldots,e_n) is an expression on A. The identity is said to hold if and only if E_1=E_2 for all x in the underlying set of A.


A good definition would allow for:
* Undefined values outside the underlying set of A, where the identity holds iff both sides are defined and equal
* Parameterized unknowns

[LAVRANOS]

GRGreathouse where did you get that definition please tell me

[LAVRANOS]

dx when your girlfriend tells that you are beautiful do you ask her to define beautiful?
You gonna loose her for ever

[dx]

dx when your girlfriend tells that you are beautiful do you ask her to define beautiful?
You gonna loose her for ever

I'm not sure whether that's a joke or not. I never debate the meaning of "identity" with my girlfriend. Also, you're not my girlfriend, so I can ask you to define beautiful :)

[LAVRANOS]

You wonna bet that you cannot define beautiful??
Why i always ask my girlfriend for an ID JUST to make sure

[Defennder]

This is getting irrelevant and personal. I'll just add in my 2 cents worth about the original question.

The trigo identities are an identity as far as mathematical theorems and proofs are concerned. a^2 + b^c = c^2, pythagoras theorem is also an identity as it can be proven.

Same goes for lots of other mathematical equations to name just a few random ones:

e^{i\pi} = -1
e^{i\theta} = \cos \theta - i \sin \theta
\vec{a} \cdot \vec{b} = |a||b| \cos \theta

All these are identities. So if they count as identities, why don't trigo identities count as identities?

[LAVRANOS]

What is a mathematical equation.
Centairly a^2 +b^2 =c^2 is an ID,but what is e^iπ=-1?

[LAVRANOS]

certainly,sorry

[Defennder]

What is a mathematical equation.
Centairly a^2 +b^2 =c^2 is an ID,but what is e^iπ=-1?The symbol there is pi. You'll learn more about this in complex numbers. It's known as Euler's identity (http://en.wikipedia.org/wiki/Euler%27s_identity).

[Count Iblis]

You wonna bet that you cannot define beautiful??
Why i always ask my girlfriend for an ID JUST to make sure

Clearly for any given neural network it is well defined when the output for a given input is "beautiful", "ugly", "abominable" etc. :biggrin:

[LAVRANOS]

Sorry i wanted to say as well that i am going to give adefinition of an identity

[dx]

When you write and i quote:A trig identity is a PROPOSITION involving = and trigonometric FUNCTIONS that is true for all values of x ,is THAT A FACT KNOWN TO EVERYONE????

That was an attempt at a definition. CRGreathouse has given you another more sophisticated attempt. I ask again, what's your idea of identity? The fact known to everyone is that \tan^2 x + 1 = \sec^2 x is an identity.


Ι BELIEVE i gave you the definition of proof and DID APPLY THE DEFINITION to couple of examples in another thread you welcome to fight it there.

But I didn't ask you for a definition of proof. I asked you for a definition of identity.

[dx]

Let me put it very specifically. You claim that \tan^{2}{x} + 1 = \sec^{2}{x} is not an identity.

Here is the question : Why is it not an identity?

In your next post, please answer this question.

[Hurkyl]

Also when i put into my computer ,tan(90),tan270,tan450,tan630 e.t.c and i mean 90 ,270,450,630 deegres i get error which means that the trig (tanx)^2+1=(secx)^2 is not true for all the values of x
(tan purple)^2 + 1 = (sec purple)^2 is not true either, but that (and your observation) are both irrelevant -- in this identity, as usually formulated, the range of x is the set of all real numbers that are not of the form 1 + 2 n pi.

This particular equation is even stronger than that -- in addition to being an identity on the range of x, it also expresses the fact the two functions have the same domain: one side of the equation is defined if and only if the other side is defined.


The equation can be tested in other contexts too -- if we take the usual extensions of the two sides of the equation to be complex meromorphic functions with domain the set of all complex numbers (and we let x range over the entire domain), then we again have an identity.

(Unlike the real versions of these functions, these functions are defined at (1 + 2 n pi) and have value equal to complex projective infinity. There are also real projective versions of these functions)

[HallsofIvy]

dx ,i asked you whether the sentance < The relation (sinx)^2 +(cosx)^2 =1 is beautiful > is atrig identity and you said yes .
Do still insist on that?
Also trhere is a great difference BETWEEN the words VALID and TRUE .VALID is an argument ,adeductuion alogical implication while TRUE OR FALSE IS a proposition or a sentance.
SO we can have a VALID argument with FALSE result.
HOW would you define an IDENTITY WITHIN logic(sympolic logic)????

You see, the difficulty is that there were SOME people here who thought you were asking a serious question and not just being a troll. Thanks for clearing that up.