sin(5x)/sin(4x) = ?
How do I cancel them out to get a constant?
sin(5x)=sin(4x+x)=2sin4xsinx
what property is this? in terms of a and b?
not true Bob,
sin(4x+x)sin4x = (sin4xcosx + cos4xsinx)sin4x
= cosx + cos4xsinx/sin4x
now sin4x = sin(3x+x) = sin3xcosx + cos3xsinx
and cos4x = cos3xcosx - sin3xsinx)
now in your list of trig expansions you might have seen:
sin3x = 3sinx - 4sin^3 x
cos3x = 4cos^3 x - 3cosx
now if you have the patience to plug all that back into the above, you will have a great mess
but all expressed in x.
Don't really know what you wanted here.
sin(4x+x)sin4x = (sin4xcosx + cos4xsinx)sin4x
should have been:
sin(4x+x) / sin4x = (sin4xcosx + cos4xsinx) / sin4x
To cancel out the terms sin(5x) and sin(4x) and obtain a constant, you need to use a trigonometric identity called the angle sum identity. The angle sum identity states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
To apply the angle sum identity to our problem, we can rewrite the numerator sin(5x) as sin(4x + x). Using the angle sum identity, we have:
sin(4x + x) = sin(4x)cos(x) + cos(4x)sin(x)
Now we can divide the numerator and denominator of the original expression by sin(4x). This gives us:
sin(5x) / sin(4x) = [sin(4x)cos(x) + cos(4x)sin(x)] / sin(4x)
Next, we can simplify the expression by canceling out the common factor sin(4x) in the numerator and denominator:
sin(5x) / sin(4x) = cos(x) + cos(4x)sin(x) / sin(4x)
At this point, we can see that sin(4x) / sin(4x) equals 1, so we can simplify the expression further:
sin(5x) / sin(4x) = cos(x) + cos(4x)sin(x)
Therefore, sin(5x) / sin(4x) cannot be simplified to a constant. It remains as cos(x) + cos(4x)sin(x).