The expression 4 sin x cos x is equivalent to which of the following?
(Note: sin (x+y) = sin x cos y + cos x sin y)
F. 2 sin 2x
G. 2 cos 2x
H. 2 sin 4x
J. 8 sin 2x
K. 8 cos 2x
Can someone please explain how to do this problem to me?
Given:
sin(x+y)=sin(x)cos(y) + cos(x)sin(y)
For the case y=x:
sin(x+x) = sin(x)cos(x) + cos(x)sin(x)
=sin(2x) = 2sin(x)cos(x)
or, just to make it easier
2sin(x)cos(x) = sin(2x)
so, what does 4sin(x)cos(x) = ?
To simplify the expression 4 sin x cos x, we can use the double-angle formula for sine:
sin 2x = 2 sin x cos x
Notice that our expression looks similar to sin 2x, but we have a coefficient of 4 instead of 2.
To get our expression to match the form of sin 2x, we can divide both sides by 2:
(4 sin x cos x) / 2 = 2 sin x cos x
Now, the expression is equivalent to 2 sin x cos x.
Comparing this simplified expression to the given options, we can see that it matches option G.
Therefore, the expression 4 sin x cos x is equivalent to 2 cos 2x.
So, the correct answer is G.
To simplify the expression 4 sin x cos x, we can use the trigonometric identity sin 2x = 2 sin x cos x.
Let's start by using this identity to rewrite 4 sin x cos x:
4 sin x cos x = 2 * 2 sin x cos x
Now, using the identity sin 2x = 2 sin x cos x, we have:
4 sin x cos x = 2 * sin 2x
So, the simplified expression is 2 sin 2x.
Therefore, the correct answer is option F. 2 sin 2x.