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

1. Apr 21, 2010

### liquidwater

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

1. The problem statement, all variables and given/known data
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)

2. Relevant equations

3. 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!

2. Apr 21, 2010

### 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.

3. Apr 21, 2010

### liquidwater

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

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

4. Apr 21, 2010

### aim1732

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

5. Apr 21, 2010

### 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.

6. Apr 21, 2010

### 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.

7. Apr 21, 2010

### 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)

8. Apr 22, 2010

### 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.

9. Apr 22, 2010

### liquidwater

Alright, thank you very much!