Question

Turn into a product:

(4sinleft(a/2right) cdot cosleft(a/2right) cdot cos 3a + sin 2a)
(A) (4sinleft(a/2right) cdot cosleft(a/2right) cdot cos 3a + sin 2a)
(B) (4sin a cdot cos a cdot cos 3a + sin 2a)
(C) (4cos^2left(a/2right) cdot cos 3a + sin 2a)
(D) (4sin a cdot cos 3a + sin 2a)

Answer 1

The product is (4sin(a/2) * cos(a/2) * (cos 3a + 2sin(a)cos(a))).

Step-by-step explanation:

The given expression is (4sin(a/2) * cos(a/2) * cos 3a + sin 2a). We can use the trigonometric identities to simplify the expression. Let's break it down step by step:

1. Using the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b), we can rewrite the expression as (4sin(a/2) * cos(a/2) * cos 3a) + (4sin(a/2) * cos(a/2) * sin 2a).
2. Using the identity sin 2a = 2sin(a)cos(a), we can simplify the second term to (4sin(a/2) * cos(a/2) * 2sin(a)cos(a)).
3. Combine the terms and simplify further: (4sin(a/2) * cos(a/2) * cos 3a + 8sin(a/2) * cos(a/2) * sin(a)cos(a)).
4. Factor out the common terms: 4sin(a/2) * cos(a/2) * (cos 3a + 2sin(a)cos(a)).

So the simplified product is (4sin(a/2) * cos(a/2) * (cos 3a + 2sin(a)cos(a))).