What Do Trigonometric Identities Really Mean and How Do You Interpret Them?

[adeserre]

I've ask around, and to my surprise, not many people knew what trigonometric identities stood for, and what they really meant. It may seem sutpidly easy to some, but it can also be challenging to others. For example, people (meaning people with an education, at my level, calc 1...)seemed to be embarrassed when they couldn't even come close to explaining what basic identities such as sine and cosine meant in a few equations. So I ask it to the community here, just to get another perspective, hopefully fresh and insightful; what do trigonometric identities such as the one pictured below mean?

In this equation, what exactly does "arctan" mean to you? How do you read it in your head? I could also add the meaning of Pi in there. Now I know Pi to be 180, but again, how do you read it in your head? The closest answer that I managed to get was(from supposedly good students...haha): "It's just a function, but who cares? Just do the math and you're done."

Sadly, this translates into: I don't need to understand the homework, I just need to complete it. hahaha

 

[HallsofIvy]

First that's not an identity, it is an integral that, presumably, can be done by using identities. Second, pi is not "180", it is (approximately) 3.1415926... I guess you meant that pi radians is equivalent to 180 degrees but that's like saying "1= 3" when you mean there are 3 feet in a yard.

What does "arctan" mean to me? The inverse function to "tan". Of course, I realize that since tangent is periodic it doesn't have a true inverse but if we restrict it to $-\pi/2$ to [/itex]\pi/2[/itex], then it does: arctan(x) returns the value y, between $-\pi/2$ and [/itex]\pi/2[/itex], such that tan(y)= x. I wouldn't say it is "just a function" but I might say- "Any good trig or precalculus book has a definition; go look it up."

Now, where did you get that formula for integration? Did you understand what substitution was made to get it?

 

[adeserre]

Haha, yes, I meant it in radians sorry, I know that pi=3.14 . As for my example of arctan, I just used the first equation that I could find containing both, regardless of the real meaning in that equation. It was more of an example, to elaborate. So I could ask again...

What would something as cos(-2x + 2y - pi) + cos( 2x - 2y - pi) mean to you?...Regardless of what followed, how do you understand it in your head? Is it simply something like tanx=sinx/cosx?(I know, cos(-2x...)is not equal to tanx lol, just an example.) Or do you rather understand it and find something that explains what dividing sinx by cosx means?

Elaborate

And thank you for the reply HallsofIvy, I tried being more specific with this post...

 

[03myersd]

To me it is just a way of finding a ratio between two numbers. I know it is not the most accurate definition but it works in my mind.

 

[DyslexicHobo]

I had a calculus 2 professor who showed proofs for all of the trigonometric substitutions used in that calculus class. I think it was one of the most useful and insightful things that can be taught. Not only does it make it more interesting when you get to see something so complex be solved so elegantly, but it really makes it make a lot more sense.

 

[adeserre]

Sadly, people just accept things such as sin/cos without any further thinking, and then go on. For example, and that is, a very sad one, when you start learning about the circumference and area of a circle, teachers often skip the foundations..."Pi=3.14...it works, just use it." hahaha...

 

[DyslexicHobo]

Another thing that I found very interesting was when I derived pi using calculus. If you find the arclength of a circle of radius 1, you can calculate pi! Our teacher didn't teach us this, but it was an interesting self-discovery. I always love it when I make connections that our teacher doesn't specifically point out. It still happens now in Calc 3. I think this is one of the most important things to show that you're actually learning the math rather than learning how to repeat certain steps to come to a solution.

 

[HallsofIvy]

So, DyslexicHobo, you are saying you "love" precisely the things adeserre is sad about!

Adserre, I have NEVER heard a teacher say "pi= 3.14... it works, just use it". Every teacher I have heard mention "pi" gave a detailed a a definition as was appropriate for the class. Similarly with trig functions. When I took a high school trig class, both the teacher and the textbook made it very clear what they meant. I'm sorry if you have had a bad experience but I think you are overstating the situation when you say "not many people knew what trigonometric identities stood for". I suspect it was more a case of people not grasping what you were asking- particularly if you spoke of " basic identities such as sine and cosine" as you do here. Since those are NOT identities, I can certainly see the possiblity that the people you were speaking to had no idea what you meant.

 

[adeserre]

Well, it seems to be a shared experience in my area, since most people do not know the uses of all these numbers. What is sine? "Just a word, who cares." could be a typical answer. "Since those are NOT identities, I can certainly see the possiblity that the people you were speaking to had no idea what you meant. ", sine and cosine not identities? I was taught otherwise, and unless I'm being absent minded, wikipedia does too...

 

[gmax137]

Well, it seems to be a shared experience in my area, since most people do not know the uses of all these numbers. What is sine? "Just a word, who cares." could be a typical answer. "Since those are NOT identities, I can certainly see the possiblity that the people you were speaking to had no idea what you meant. ", sine and cosine not identities? I was taught otherwise, and unless I'm being absent minded, wikipedia does too...

Is this a language barrier issue? Maybe adessere means "entities" rather than "identities." The confusion would arise, due to our understanding that things like sin^2 + cos^2 = 1 are called "trigonometric identities."

just a thought

 

[DyslexicHobo]

So, DyslexicHobo, you are saying you "love" precisely the things adeserre is sad about!

Well maybe I worded that poorly. I don't like the fact that I have to be the one that discovers connections on my own, rather than having the teacher point them out like they should. I just love the feeling that I understand the fundamentals at a higher level when I am able to make such connections.

 

[lurflurf]

Exactly, teachers should not point out obvious things. Students should understand fundamentals and make such connections for themselves.

 

[Redbelly98]

In this equation, what exactly does "arctan" mean to you? How do you read it in your head?

It means the angle whose tangent is ____. I think of either that, or I think of a graph of the function, or I picture a right triangle (and the appropriate angle and sides) in my head. Depending on the nature of a specific problem, I'll probably focus on one of those meanings to work through it.

 

[Mentallic]

... If you find the arclength of a circle of radius 1, you can calculate pi! Our teacher didn't teach us this, but it was an interesting self-discovery...

lol isn't this the definition of $$C=2\pi r$$?

I don't feel it's the teacher's job to spoon-feed you all the fundamentals, but rather teach you the topic thoroughly, giving you the knowledge to be able to figure this stuff out for yourself.

 

[daniel_i_l]

Another thing that I found very interesting was when I derived pi using calculus. If you find the arclength of a circle of radius 1, you can calculate pi!

It's impossible to "calculate pi". You can get an arbitrarily good approximation though.