Express sin4x in terms of sinx?

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In summary: I can use cos(2x) = cos^2(x) - sin^2(x), and sin(2x) = 2sin(x)cos(x), to say that sin(4x) = 2sin(2x)cos(2x) = 2(2sin(x)cos(x))(cos^2(x) - sin^2(x)) = 4sin(x)cos(x)(1-sin^2(x)) = 4sin(x)cos(x)-4sin^3(x).In summary, sin(4x) can be expressed in terms of sin(x) as 4sin(x)cos(x)-4sin^
  • #1
kgh0st
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express sin4x in terms of sinx?

Alright so I am working on my homework tonight (i'm in trig/calc), and I get everything done but the last problem. Anyways, I've been working on this for a while now and I can't even get an idea of where to start. Anyways, he says we need to express sin4x in terms of sinx. Can anyone help me out as to how to do this?

Thanks in advance.
 
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  • #2
HINT:

[tex]\sin 4x = 2 \sin 2x \cos 2x[/tex]
 
  • #3
kgh0st said:
Alright so I am working on my homework tonight (i'm in trig/calc), and I get everything done but the last problem. Anyways, I've been working on this for a while now and I can't even get an idea of where to start. Anyways, he says we need to express sin4x in terms of sinx. Can anyone help me out as to how to do this?

Thanks in advance.
I don't think sin(4x) can be expressed in terms of only sin(x). It can, however be expressed in terms of sin(x), and cos(x).
In general, sin(2nx), where n is a natural number that's greater than or equal to 1, can be expressed in terms of sin(x), and cos(x), but not sin(x) alone.
sin((2n + 1)x), n >= 0, however can be expressed in terms of only sin(x).
 
  • #4
Well, actually, it can, since cos(x) = +sqrt(1-sin^2(x))
 

1. What is the formula for expressing sin4x in terms of sinx?

The formula for expressing sin4x in terms of sinx is 4sinx cosx - 8sin^3x cosx + 4sin^5x.

2. How do I simplify an expression involving sin4x in terms of sinx?

To simplify an expression involving sin4x in terms of sinx, use the trigonometric identity sin2x = 2sinx cosx to rewrite sin4x as 2sin2x cos2x. Then, use another trigonometric identity, sin2x = 2sinx cosx, to simplify further to 4sinx cosx - 8sin^3x cosx + 4sin^5x.

3. Can I express sin4x as a power of sinx?

Yes, sin4x can be expressed as a power of sinx. Using the double angle formula, sin2x = 2sinx cosx, sin4x can be rewritten as 2sin2x cos2x, which can then be further simplified to 4sinx cosx - 8sin^3x cosx + 4sin^5x.

4. Is there a shortcut to expressing sin4x in terms of sinx?

Yes, there is a shortcut to expressing sin4x in terms of sinx. You can use the trigonometric identity sin4x = 4sinx cosx - 8sin^3x cosx + 4sin^5x, which can be derived from the double angle formula and simplification using trigonometric identities.

5. How can I use the expression sin4x in terms of sinx to solve trigonometric equations?

To solve trigonometric equations involving sin4x, you can use the expression sin4x = 4sinx cosx - 8sin^3x cosx + 4sin^5x. This can help you simplify the equation and solve for x by factoring or using other algebraic methods.

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