Use '3 sin x - 4(sin x)^3' to show that if sin x = sin (3x), then sin x = 0 or

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In summary, the conversation discusses using trigonometric formulas to find an expression for sin(3x) in terms of sin x alone. It then asks to show that if sin x = sin(3x), then sin x is either 0 or +/- 1/sqrt(2). The conversation also touches on the idea of eliminating sin3x from the equation to solve for sinx, but it is important to note that this cannot be done algebraically as sine expands by rules of trigonometry, not algebra.
  • #1
liquidwater
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Use '3 sin x - 4(sin x)^3' to show that if sin x = sin (3x), then sin x = 0 or...

Homework Statement


Use double-angle and addition formulæ and other relations for trigonometrical
functions to find an expression for sin(3x) in terms of sin x alone. Use the expression you
have found to show that, if sin x = sin (3x) , then either sin x = 0 or sin x = (+ or -)1/sqrt(2)


Homework Equations



The Attempt at a Solution



I have found that [tex]sin(3x) = 3 sin x - 4 sin^3 x[/tex] (I think).

However, I do not know what is meant by the latter part of the question. Would love for someone to explain it to me!
 
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  • #2
It says that you should equate sin3x and sinx in the equation you found out and then solve the equation as a polynomial in sinx.
 
  • #3
Solve [tex]-4 sin ^3 x + 3 sin x = sin 3x[/tex] and [tex]-4 sin ^3 x + 3 sin x = sin x[/tex] ?

edit: or Solve [tex]-4 sin ^3 3x + 3 sin 3x = 0[/tex] ?
 
  • #4
Doesn't matter which one. It is like replacing x with y in an equation and saying x=y.
 
  • #5
The three equations I wrote are totally different aren't they...

one is y = sin(3x), another y = sin(x) and the other y = 0. (where y is the same equation). I think I misunderstood your first post.
 
  • #6
Eliminate sinx or sin3x to get a value for either. Since sinx=sin3x you would get same value for both and it shouldn't matter as such, although the question intends you to eliminate sin3x.
 
  • #7
"Eliminate" sin3x from [tex]sin(3x) = 3 sin x - 4 sin^3 x[/tex]?

Sorry for not understanding but I'm really lost :(.

Do I solve [tex]sin(x) = 3 sin x - 4 sin^3 x[/tex]?And how does sin x = sin 3x when sin(1) isn't equal to sin(3)
 
  • #8
The second equation you wrote was exactly what I was referring to.
And if I am right you are cancelling x from sin x =sin 3x to get sin1 = sin3 ? If that's the case then you should know that you cannot do anything with what is inside the argument of sine, it expands by rules of trigonometry and not algebra.
 
  • #9
aim1732 said:
The second equation you wrote was exactly what I was referring to.
And if I am right you are cancelling x from sin x =sin 3x to get sin1 = sin3 ? If that's the case then you should know that you cannot do anything with what is inside the argument of sine, it expands by rules of trigonometry and not algebra.

Alright, thank you very much!
 

1. What is the equation 3 sin x - 4(sin x)^3 used for?

The equation 3 sin x - 4(sin x)^3 is used to show the relationship between the sine of an angle and its tripled angle.

2. How does this equation relate to the identity sin x = sin (3x)?

By substituting 3x for x in the equation 3 sin x - 4(sin x)^3, we get 3 sin (3x) - 4(sin (3x))^3. This is equivalent to the identity sin x = sin (3x).

3. What does this equation reveal about the values of sin x and sin (3x)?

The equation shows that if sin x = sin (3x), then either sin x = 0 or sin (3x) = 0. This means that the values of sin x and sin (3x) are equal only when one of them is equal to 0.

4. Can this equation be used to solve for the values of x?

No, this equation alone cannot be used to solve for the values of x. It only shows the relationship between sin x and sin (3x) when they are equal.

5. Can this equation be used to prove other trigonometric identities?

Yes, this equation can be used to prove other identities involving the sine function, such as the double angle identity sin 2x = 2sin x cos x.

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