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I can't get the right answer for my math question

Posted 10 years ago

Hey all

I have been stuck with a question and i tried to see if wolframalpha could help me. The question is: simplify and develop sin (x+55°)

I typed that into the search box but it gives me the wrong answer. The right answer in the book are sin x cos(55°) + cos(x) sin(55°).

Why doesn't it show me the right answer? Have i typed in the wrong command.

I have bought the app

Thanks Farah

POSTED BY: Nicklas Karlsson
9 Replies
Posted 10 years ago

Hi Udo, Wittgenstein is my favorite philosopher! And I think very apropos to this discussion. He called philosophy "thinking befuddled by language," and told us that the most difficult answers to find were the answers to invalid or ill-defined questions. By eliminating such questions, he came to the conclusion that he had solved all the real questions in philosophy, and was amazed at how little he had done.

In the very first post, Farah says, "The question is: simplify and develop sin (x+55°)." To ask anyone or any tool to answer that requires a clear definition. Bruce points out that sin(a+b) is simpler than sin(a)cos(b)+cos(a)sin(b). That is certainly true by the most obvious criteria, but probably not what the trig teacher had in mind. I suspect that a better statement of the intended problem would be, "find an equivalent expression in which all arguments appear as atomic forms." Perhaps there is a way to tell Mathematica that in its repeated evaluations that would be a simpler and irreducible form.

Best, David

POSTED BY: David Keith

Hi David,

for completeness here the quotes and the reference, for sure you have it on your book shelf or do find it easily in English

66. Die Prophezeiung lautet nicht, daß der Mensch, wenn er bei der Transformation dieser Regel folgt, das herausbringen wird - sondern, daß er, wenn wir sagen, er folge der Regel, das herausbringen werde.

...snip...

67. Zum Rechnen gehört wesentlich dieser Consensus, das ist sicher. D.h.: zum Phänomen unseres Rechnens gehört dieser Consensus.

In einer Rechentechnik müssen Prophezeiungen möglich sein. Und das macht die Rechentechnik der Technik eines Spieles, wie des Schachs, ähnlich.

Aber wie ist das mit dem Consensus - heißt das nicht, daß ein Mensch allein nicht rechnen könnte? Nun, ein Mensch könnte jedenfalls nicht nur einmal in seinem Leben rechnen.

...snip...


Ludwig Wittgenstein, Werkausgabe Band 6

Bemerkungen über die Grundlagen der Mathematik

Herausgegeben von G. E. M. Anscombe, Rush Rhees, G. H. von Wright

Suhrkamp Taschenbuch Wissenchaft, Bd. 506, 6. Auflage 1999

Teil III 1939 - 1940, S. 192 - 193

POSTED BY: Udo Krause
Posted 10 years ago

Generally, the problems you are discussing involve working with trig identities. These are part of the core learning in a trigonometry class. They are most useful when manipulating trig expressions into a form which, for some reason, makes the underlying structure of the model more evident. Most often, this is something being done (please forgive me) with pencil and paper. The reason for this is that, while Mathematica is a marvelous tool, all such tools still fall far short of the capabilities of the human mind.

I would guess that one reason for that lies in the words "more evident." In working physics problems, for example, those who have done it for a long time (and some right away) develop an intuition that tells them when a particular rearrangement of an expression is moving toward a more useful form. They know they are trying to get to a form which will make certain characteristics of the model evident. An example might be the derivation of a wave equation from Maxwell's equations. He knew immediately that what he had was astounding.

POSTED BY: David Keith

Most often, this is something being done (please forgive me) with pencil and paper.

No pencil, no paper in the future of pure mathematics!


There is another question - a Wittgenstein-style [3] question - can mathematics be learned without pencil and paper? How do you come to an understanding what needs to be done to solve something? All too often - in this forum too - posters do not understand why Mathematica does not just solve it! Maybe there is some mathematical reality preventing it from doing so. Maybe the job has some meaning, forcing the human problem solver to reformulate it and then to use some more or less educated combinations of Mathematica commands, usually called a program.

Wolfram Inc. itself decided to come half-way to the customer, look at incredible things like the FrenetSerretSystem or the FEM operator and boundary condition stuff they built in. The customer just calls that ...


[3] Wittgenstein asked: Can a single man calculate or even count? I.e. a human being without a society or community surrounding her/him.

POSTED BY: Udo Krause

First look at a plot

Plot[Sin[x]^2 - Cos[x]^2, {x, -2 Pi, 2 Pi}]

to see the solution, then compute it

In[6]:= Reduce[Sin[x]^2 - Cos[x]^2 == 1, x]
Out[6]= C[1] \[Element]  Integers && (x == -(\[Pi]/2) + 2 \[Pi] C[1] || x == \[Pi]/2 + 2 \[Pi] C[1])

if you don't trust your eyes.

POSTED BY: Udo Krause

Hi,

I guess that the function you are looking for is TrigExpand. This will give you (nearly) the required result:

http://www.wolframalpha.com/input/?i=TrigExpand%5BSin%5Bx+%2B+55%5D%5D

which the drawback that Wolfram Alpha does not understand that the 55 is in degrees. You can also use

http://www.wolframalpha.com/input/?i=TrigExpand%5BSin%5Bx+%2B+55+degrees%5D%5D

which understands that 55 is in degrees, but then it numerically evaluates the sin and cos of the 55 degrees, so that it looks a bit different.

May I suggest that you sign up for a (free) Wolfram Programming account. Then you have access to the full Wolfram language and this command will do the trick:

TrigExpand[Sin[x + HoldForm[55 \[Degree]] ]]

Cheers,

Marco

POSTED BY: Marco Thiel

Thanks for the answer, but how should i know the next time if i have a problem? must i learn the wolframalpha langue that you suggested?

I tested with a new question sin^2(x)- cos^2(x) =1 in the book it says the answer are x = 90°+ n x 360° . How should i think here? Iam really sorry if iam bothering you.

POSTED BY: Nicklas Karlsson

Dear Farah,

I understand that it can be confusing at first to solve all these different problems, using slightly different functions/commands. As Udo Krause suggested you could use Reduce for this second problem. You can also use Solve:

Solve[Sin[x]^2 - Cos[x]^2 == 1, x]

This gives the result in radian, but it can be easily converted to Degree using another command in the Wolfram Language:

Needs["Units`"]
Solve[Sin[x]^2 - Cos[x]^2 == 1, x]
Convert[Evaluate[x /. %[[1]]], Degree]

It might look a bit intimidating at first, but it is actually quite simple. These commands build up, i.e. you nest them and the expressions become larger and to the untrained eye difficult to decipher.

You ask whether you should rather use learn the Wolfram Language. I personally would strongly suggest to learn it. It is much more flexible than just using WolframAlpha, which is of course a powerful tool, too. If you learn the Wolfram Language you will soon be able to "ask" complicated questions and you will have a fantastic tool to support your learning of Mathematics. I don't think that knowing how to use Mathematica can substitute learning and understanding mathematics, but it certainly can support your learning a great deal. You can check results of problems you solved or sometimes even get step-by-step solutions. But you can do so much more. Very often when you learn mathematics the problems in textbooks are rather dull and it feels like a chore to have to solve them. Often that is because the really interesting problems are much too complicated to solve with paper and pencil - some very advanced mathematicians might be able to do that at least partially, but at School or even as an Undergraduate student at University you might not be able to do it. Mathematica is what you need to extend what you have learnt at School to really interesting problems. If you learn the Wolfram Language you might actually find that, after a little training, you can solve your maths problems, but also that maths can be fun. You can get a much deeper understanding of what is going on if you use mathematica to "figure stuff out". For example your first question, treats an identity to "split" a sine function into a sum of products of sine and cosine functions. Have a look at this demonstration project:

http://demonstrations.wolfram.com/AdditionFormulaForSine/

The key output looks like this:

enter image description here

By just rearranging the coloured triangles you can actually see why the result that Mathematica gives is true. Actually, if you understand that figure you can easily write down the mathematical proof, why this is true.

It is definitely worthwhile to learn Mathematica. You will still have to learn the maths, but it will be more fun.

Cheers,

Marco

PS: Here you can find some other examples of what Mathematica can do with trigonometric identities.

POSTED BY: Marco Thiel

a) I do not know what you mean by "develop" here.

b) sin(x+55°) is simpler than sin x cos(55°) + cos(x) sin(55°).

c) Is the result that Wolfram|Alpha gives you actually wrong, or just different? It gives me cos((7 pi)/36-x), which appears to be equivalent.

In[1]:= FullSimplify[ Cos[(7 Pi)/36 - x] -  Sin [x + 55 \[Degree]],  Element[x, Reals]] 
Out[1]= 0 

Can you show us the output you get, either with a screen shot, or by clicking on it, then clicking the A in the little menu that pops down, then copying the text version?

enter image description here

POSTED BY: Bruce Miller
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