Tan x

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SomeRandomLearner

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Looking to learn how tan(55) = 1.428.....

Google only takes me to calculators and I'm not interested in knowing the answer. I want to understand the process that brings about the answer.
 
SomeRandomLearner said:
Looking to learn how tan(55) = 1.428.....

Google only takes me to calculators and I'm not interested in knowing the answer. I want to understand the process that brings about the answer.
Click to expand...

Let me also say that I know for a triangle tan = O/A but to refine the question when I enter tan(55) into the calculator what is the formula that is used to provide the answer of 1.428?
 
Is there any reason you are asking about the specific angle of 55 degrees? Or are you looking for a more general algorithm that the calculator uses?

-Dan
 
There are several ways in which trig functions can be evaluated, all of which are approximations.

One way (which you could do by hand, in principle) would be to evaluate terms of an infinite series or products. Here are some references:


My understanding is that calculators typically use the CORDIC algorithm; here are a couple places I've found it explained:

en.wikipedia.org

CORDIC - Wikipedia

en.wikipedia.org en.wikipedia.org
 
Jomo said:
Why? The OP clearly wanted the general way to calculate the tangent of any value.
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C'mon, Jomo. Just trying to help. Triangles are very visual, and sometimes they can make things simpler to understand.
 
SomeRandomLearner said:
Looking to learn how tan(55) = 1.428.....

Google only takes me to calculators and I'm not interested in knowing the answer. I want to understand the process that brings about the answer.
Click to expand...
You can use Taylor series expression for tan(x) [ x in radian and |x| <= pi/2]

tan(x) = x + x^3/3 + 2x^5/15 + 17x^7/315 ......
 
firemath said:
C'mon, Jomo. Just trying to help. Triangles are very visual, and sometimes they can make things simpler to understand.
Click to expand...
The OP wanted the formula that a calculator uses to compute tangent of angles. How on earth will looking at a triangle help. C'mom firemath!
 
Jomo said:
The OP wanted the formula that a calculator uses to compute tangent of angles. How on earth will looking at a triangle help.
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To understand a trigonometric formula, it might be helpful to look at a triangle that would use that computation.
[MATH]tan=\frac{O}{A}[/MATH], so can't we look at a real example of O and A? I do not suggest that my way is the only way-- note the "might be helpful."

Jomo said:
....C'mom....
Click to expand...
?
 
The OP wanted the formula tan(x)=x+x^3/3 + 2x^5/15 + 17x^7/315+...

The formula used on a calculator would actually end based on the number of digits which a calculator uses.

I am not up on even an approximation of how terms a calculator would use.

All I can say is good luck looking at a triangle and coming up with the above formula.

Please try to learn from us. Personally I am getting a great education from this website. There are some extremely bright people here (certainly not me) that you can learn from including me.

I have never 'attacked' you unless your post was not correct.
 
Subhotosh Khan said:
You can use Taylor series expression for tan(x) [ x in radian and |x| <= pi/2]

tan(x) = x + x^3/3 + 2x^5/15 + 17x^7/315 ......
Click to expand...
I was going to suggest the series but what happens near [math]x = \pi /2[/math]? The series wouldn't be valid then, will it?

I was waiting to see what someone might suggest. I have a possibility, but it's a little convoluted.

First, calculators multiply and divide using logarithms, which are built into the machine. So my thought was
[math]sin(x) = x - \dfrac{1}{6} x^3 + \dfrac{1}{120} x^5 - \text{ ...}[/math]and
[math]cos(x) = 1 - \dfrac{1}{2} x^2 + \dfrac{1}{24} x^4 - \text{ ...}[/math]
Then
[math]ln(tan(x)) = ln(sin(x)) - ln(cos(x))[/math]
Possible?

-Dan
 
I suspect that it could be possible.

After reading what you wrote it may be possible, if not probable, that the calculator already having the
expansion for sin(x) and cos(x) that it might simply compute the tan(x) as sin(x)/cos(x). In fact, with just the expansion of sin(x) and cos(x) you can get all the trig functions. Hmm, I guess that even with just sin(x) or cos(x) you can get all trig functions.
 
As I said previously, my (third-hand) understanding is that calculators use the CORDIC algorithm, which is not a series, but a way to start with a small table and quickly refine a value from that.

In effect, it is as if they took any angle and express its tangent in terms of an angle in a smaller domain, to which they could apply the series; but it is really sort of the reverse of that.

Ultimately, you would have to ask a calculator manufacturer. It may be something a little different that they actually use.

Here are a couple more articles about it:

www.allaboutcircuits.com

An Introduction to the CORDIC Algorithm - Technical Articles

CORDIC is a hardware-efficient iterative method which uses rotations to calculate a wide range of elementary functions.
www.allaboutcircuits.com www.allaboutcircuits.com

 
I am sure that in the past that I was told that the calculator used the initial part of series for some functions (like trig functions). Was I told wrong or has things changed since then?
 
I read the article about the Queens College (CUNY!!) professor and that basically answers my question. Thank you!
 
Jomo said:
I am sure that in the past that I was told that the calculator used the initial part of series for some functions (like trig functions). Was I told wrong or has things changed since then?
Click to expand...
I think the story is just like you see a little in this thread: Often rather than look into how it is actually done, teachers use such a question as a teaching opportunity, explaining how what they teach (e.g. series) could be used. The reality is more complicated, because calculator or computer designers have slightly different needs, so their solutions will be different -- in this case, considerably more efficient but harder to explain in a classroom.

Here is what I believe is the first place I heard about CORDIC, from a math professor who was told by a math professor that it is not done (as they both would have expected) by series:


The article by Prof. Sultan says the same sort of thing, and very nicely.
 
I appreciate all the replies. The only reason I wanted to know is because I've been working with tangent via triangles and had decided to just use tan(55) and it spat out a value. So I looked up that value. I was just curious as to how the calculator came to the conclusion that tan(55) = 1.42814 when it isn't given any other values.

Also on a SAT/ACT prep site I found some information . They used a formula tan = sin/cos. Keep in mind I stopped my math at Algebra 1 and have just started needing to learn trig to calculate angles for fabrication.
 
SomeRandomLearner said:
I appreciate all the replies. The only reason I wanted to know is because I've been working with tangent via triangles and had decided to just use tan(55) and it spat out a value. So I looked up that value. I was just curious as to how the calculator came to the conclusion that tan(55) = 1.42814 when it isn't given any other values.
Click to expand...
In that case, the answer is, "magic". ;)

For a broader look at what trig functions are all about, you might be interested in my blog post about it. There I briefly mention the genie that lives inside your calculator. If you don't believe in genies, I also linked to this page that explains how the genie does it, in terms similar to what we've discussed in this thread.
 
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