Math Is Fun Forum

Al-Allo

Member

Registered: 2012-08-23

Posts: 324

Evaluating arctan(−10)

I was wondering, how could I find the exact value of arctan(−10) ? We know that an approximation of the exact value would be arctan(−10)≈−1.47112 rad but if we wanted the exact value in radians, How would I find it ?

Thank you!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Evaluating arctan(−10)

Hi;

One way to calculate to as many digits as is necessary is to make use of the identity:

Now we use the Taylor series:

substituting x = 10 in the RHS we get 1.471127674302833 which is off by one in the 12th place.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Al-Allo

Member

Registered: 2012-08-23

Posts: 324

Re: Evaluating arctan(−10)

I didn't see taylor series... and i want a closed form for arctan(-10)...

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Evaluating arctan(−10)

A closed form? It is already a closed form.

In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, certain "well-known" operations (e.g., + − × ÷), and functions (e.g., nth root, exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but e.g. usually no limit. The set of operations and functions admitted in a closed-form expression may vary with author and context.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,974

Website

Re: Evaluating arctan(−10)

I was wondering, how could I find the exact value of arctan(−10) ? We know that an approximation of the exact value would be arctan(−10)≈−1.47112 rad but if we wanted the exact value in radians, How would I find it ?

Thank you!

arctan is already a closed form. What you are looking for is a finite combination of +,-,*,/, and ^ which is called an Algebraic Expression.

Why must arctan(-10) have an Algebraic Expression? As a matter of fact it does not have such an expression.

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Al-Allo

Member

Registered: 2012-08-23

Posts: 324

Re: Evaluating arctan(−10)

Ok, but how am I supposed to know if there is an algebraic expression or not ? Whether we're talking about arctan, arcsin, arccos, etc.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Evaluating arctan(−10)

In some ways that is a very difficult question to answer. In the last part of the 20th century a couple of guys starting with Ferguson developed algorithms to determine whether or not some number could be represented in terms of known constants.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,974

Website

Re: Evaluating arctan(−10)

Galois theory!

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Evaluating arctan(−10)

Nope, something way more important. It has been called the algorithm of the 20th century and hardly anyone knows anything about it.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Al-Allo

Member

Registered: 2012-08-23

Posts: 324

Re: Evaluating arctan(−10)

OK, I thought that we could always represent them in algebraic expressions....anywya, thank you!

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Evaluating arctan(−10)

Nope, something way more important. It has been called the algorithm of the 20th century and hardly anyone knows anything about it.

Might that be PSLQ?

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“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,974

Website

Re: Evaluating arctan(−10)

Or LLL

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Evaluating arctan(−10)

All true and correct.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.