simplify sin (8x) using only sin(x) and cos (x)
sin8x = 2sin4xcos4x
= 4sin2xcos2x(cos^2(2x)-sin^2(2x))
= 8sinx*cosx(cos^2(x)-sin^2(x))[(cos^2(x)-sin^2(x)]^2 - 4sin^2(x)cos^2(x)]
and you can expand that in many ways, one of which is, using s=sin(x) c=cos(x):
8sc^7 - 56s^3c^5 + 56s^3c^5 - 8s^7c
Sure, here's an attempt at simplifying sin (8x) using only sin(x) and cos(x):
Step 1: Use the sine double-angle formula: sin(2x) = 2sin(x)cos(x).
sin(8x) = sin(4x + 4x)
= sin(4x)cos(4x) + cos(4x)sin(4x)
= (2sin(2x)cos(2x))(2cos(2x) - 1) + (2cos(2x) - 1)(2sin(2x)cos(2x))
= 4sin(2x)cos^2(2x) - 2sin(2x)cos(2x) + 4sin^2(2x)cos(2x) - 2sin(2x)cos(2x)
= 4sin(2x)[cos^2(2x) + sin^2(2x)] - 4sin(2x)cos(2x)
= 4sin(2x) - 4sin(2x)cos(2x)
Step 2: Simplify sin(2x) using the double-angle formula.
sin(2x) = 2sin(x)cos(x)
Substituting this back into the expression for sin(8x):
sin(8x) = 4(2sin(x)cos(x)) - 4(2sin(x)cos(x))
= 8sin(x)cos(x) - 8sin(x)cos(x)
= 0
Therefore, sin(8x) simplifies to 0 using only sin(x) and cos(x).
To simplify sin(8x) using only sin(x) and cos(x), we can use the multiple angle formula for sin(2x).
The multiple angle formula for sin(2x) is:
sin(2x) = 2 * sin(x) * cos(x)
We can rewrite sin(8x) as sin(4x + 4x). Applying the multiple angle formula twice, we get:
sin(8x) = 2 * sin(4x) * cos(4x)
= 2 * [2 * sin(2x) * cos(2x)] * [2 * cos^2(2x) - 1]
= 4 * sin(2x) * [2 * cos^2(2x) - 1]
Now, let's substitute sin(2x) using another multiple angle formula:
sin(2x) = 2 * sin(x) * cos(x)
Substituting the value into our equation, we have:
sin(8x) = 4 * [2 * sin(x) * cos(x)] * [2 * cos^2(2x) - 1]
simplifying this expression further would not be possible using only sin(x) and cos(x) since it involves cos^2(2x).
To simplify sin(8x) using only sin(x) and cos(x), we can use the multiple-angle formula for sine.
The multiple-angle formula for sine states that sin(nx) can be expressed in terms of sin(x) and cos(x) as follows:
sin(nx) = sin(x) * cos((n-1)x) + cos(x) * sin((n-1)x)
In this case, we want to simplify sin(8x). To do so, we can use the multiple-angle formula for sine with n = 8.
sin(8x) = sin(x) * cos(7x) + cos(x) * sin(7x)
Notice that we have now expressed sin(8x) using sin(x) and cos(x) only.