How do you express sin4x as a trigonometric function of x?
use angle identities
sin 2a = 2 sin a cos a
cos 2a = co^2a - sin^2 a
use them once for a = 2x
then again for a = x
To express sin(4x) as a trigonometric function of 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 express sin(4x) in terms of x, so we substitute n = 4 in the formula:
sin(4x) = sin(x) * cos(3x) + cos(x) * sin(3x)
Now, let's expand the cosine and sine terms further using the double-angle formulas:
cos(3x) = cos(2x + x)
= cos(2x) * cos(x) - sin(2x) * sin(x)
sin(3x) = sin(2x + x)
= sin(2x) * cos(x) + cos(2x) * sin(x)
Now, substitute these expressions into the equation for sin(4x):
sin(4x) = sin(x) * [cos(2x) * cos(x) - sin(2x) * sin(x)] + cos(x) * [sin(2x) * cos(x) + cos(2x) * sin(x)]
Continuing to simplify:
sin(4x) = sin(x) * cos(2x) * cos(x) - sin(x) * sin(2x) * sin(x) + cos(x) * sin(2x) * cos(x) + cos(x) * cos(2x) * sin(x)
Further simplifying:
sin(4x) = sin(x) * cos(2x) * cos(x) + cos(x) * sin(2x) * cos(x) - sin(x) * sin(2x) * sin(x) + cos(x) * cos(2x) * sin(x)
Now, let's rearrange the terms and factor out sin(x) to express sin(4x) as a trigonometric function of x:
sin(4x) = sin(x) * [cos(2x)*cos(x) + sin(2x)*cos(x) - sin(2x)*sin(x) + cos(2x)*sin(x)]
Finally, combining like terms:
sin(4x) = sin(x) * [cos(2x + x) + sin(2x + x)]
We can rewrite the expression inside the brackets as a single trigonometric function using the sum-to-product formulas:
sin(4x) = sin(x) * sin(3x + x)
sin(4x) = sin(x) * sin(4x)
Thus, sin(4x) can be expressed as sin(4x) itself in terms of x.