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

[liquidwater]

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 $$sin(3x) = 3 sin x - 4 sin^3 x$$ (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!

 

[aim1732]

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.

 

[liquidwater]

Solve $$-4 sin ^3 x + 3 sin x = sin 3x$$ and $$-4 sin ^3 x + 3 sin x = sin x$$ ?

edit: or Solve $$-4 sin ^3 3x + 3 sin 3x = 0$$ ?

 

[aim1732]

Doesn't matter which one. It is like replacing x with y in an equation and saying x=y.

 

[liquidwater]

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.

 

[aim1732]

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.

 

[liquidwater]

"Eliminate" sin3x from $$sin(3x) = 3 sin x - 4 sin^3 x$$?

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

Do I solve $$sin(x) = 3 sin x - 4 sin^3 x$$?And how does sin x = sin 3x when sin(1) isn't equal to sin(3)

 

[aim1732]

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.

 

[liquidwater]

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!