How to determine whether two complex trig equation is identical.

In summary, the Integral machine does not always output the same answer as our answers. This can be difficult to compare, as the answer in the second example has an argument in it that is difficult to simplify.
  • #1
PrudensOptimus
641
0
I believe, many of us had this problem:


After finding the integral of some function, we wonder, what the right answer would be. So we goto http://integrals.wolfram.com to look up the answer. The answer the "Integral machine" gave is not always in the same form our answers are.


For example: ∫ Cos[x]^3 = u - u^3/3, where u = sin x.

The answer the integral machine outputs would be something more nastier...


Can someone explain, how do you generally determine whether an expression is identical to another expression(usually in a more complex form)?
 
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  • #2
I don't know of a general method, but you can handle trigonometric cases by using the relevant identities. For instance, in the example you cited you have:

∫cos3(x)dx=sin(x)-(1/3)sin3(x)+C

Wolfram's "Integral Machine" gave this answer:

∫cos3(x}dx=(3/4)sin(x)+(1/12)sin(3x)+C

(note, in both cases I added the "+C" in myself).

The obvious difficulty in comparing the above antiderivatives is in the "3x" argument in the second one. You need to use an identity that reduces the argument to "x" in every term.

The identity is:

sin3(x)=(3/4)sin(x)-(1/4)sin(3x)
 
  • #3
isn't Sin3x also ((1-cos2x)/2)*sinx?
 
  • #4
Originally posted by PrudensOptimus
isn't Sin3x also ((1-cos2x)/2)*sinx?

No, sin2(x)=(1/2)(1-cos(2x)).
 
  • #5
one way that will at least disprove the two formula's equality would be plugging in a few numbers.

i tried to see if the integrals site will accept the command FullSimplify or Simplify in the integrand. first of all, to see if that would even help, i inputted 0 for the integrand. got nothing. so i inputted 1, for which i got x. the idea would have been to have it integrate 1+FullSimplify[G(x)-H(X)] where G and H are the two things you want to see if are equal. if the answer was x, then they're equal.

i had it integrate 1+(Sin[x]^2)-((1-Cos[2x])/2) and it gave x as the answer, so it seems to Simplify or FullSimplify its answers before drawing them.

so to see if H(x)=G(x), have it integrate 1+H(x)-G(x) and if it gives x then H(x)=G(x). having it integrate H(x)-G(x) to get 0 has worked at least once. the problem with doing that is if it thinks H(x)-G(x) is 0 prior to integrating, it won't like it.

it even knew that sin3(x)=(3/4)sin(x)-(1/4)sin(3x) or at least that ∫sin3(x)=∫(3/4)sin(x)-(1/4)sin(3x)
 
Last edited:

1. How do you compare two complex trigonometric equations?

To compare two complex trigonometric equations, you need to simplify each equation using trigonometric identities and then set them equal to each other. If the simplified equations are equal, then the original equations are identical.

2. What is the first step in determining if two complex trigonometric equations are identical?

The first step is to simplify both equations using trigonometric identities. This will help to reduce the complexity of the equations and make it easier to compare them.

3. How do you use trigonometric identities to simplify equations?

Trigonometric identities are equations that show the relationship between different trigonometric functions. By substituting these identities into the original equations, you can simplify them and make them easier to compare.

4. Can two complex trigonometric equations be identical even if they look different?

Yes, two complex trigonometric equations can look different but still be identical. This is because they may have different forms but can be simplified to the same equation using trigonometric identities.

5. What is the importance of determining if two complex trigonometric equations are identical?

Determining if two complex trigonometric equations are identical is important because it allows you to solve for unknown variables by using the simplified equation. It also helps to verify the accuracy of mathematical expressions and equations.

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