Question

Sin (alfa+ pi/8)cos (alfa-pi/24)

Answer 1

на фотографии всё Расписала, Надеюсь что будет понятно

Answer 2

To simplify the expression, let's use the trigonometric identity:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Applying this identity to the expression sin(alfa + pi/8)cos(alfa - pi/24), we have:

sin(alfa + pi/8)cos(alfa - pi/24) = sin(alfa)cos(pi/8)cos(alfa)cos(-pi/24) + cos(alfa)sin(pi/8)sin(alfa)cos(-pi/24)

Since cos(-x) = cos(x) and sin(-x) = -sin(x), we can simplify further:

= sin(alfa)cos(pi/8)cos(alfa)cos(pi/24) - cos(alfa)sin(pi/8)sin(alfa)cos(pi/24)

Using the identity cos(pi/8) = cos(pi/24) = sqrt(2 + sqrt(2))/2 and sin(pi/8) = sin(pi/24) = sqrt(2 - sqrt(2))/2, the expression becomes:

= sin(alfa)(sqrt(2 + sqrt(2))/2)cos(alfa)(sqrt(2 + sqrt(2))/2) - cos(alfa)(sqrt(2 - sqrt(2))/2)sin(alfa)(sqrt(2 - sqrt(2))/2)

= (sqrt(2 + sqrt(2))/2)(sqrt(2 + sqrt(2))/2)sin(alfa)cos(alfa) - (sqrt(2 - sqrt(2))/2)(sqrt(2 - sqrt(2))/2)sin(alfa)cos(alfa)

= (2 + sqrt(2))/4 sin(2alfa) - (2 - sqrt(2))/4 sin(2alfa)

= ((2 + sqrt(2)) - (2 - sqrt(2)))/4 sin(2alfa)

= (2 + sqrt(2) - 2 + sqrt(2))/4 sin(2alfa)

= (2sqrt(2))/4 sin(2alfa)

= sqrt(2)/2 sin(2alfa)

Therefore, sin(alfa + pi/8)cos(alfa - pi/24) simplifies to sqrt(2)/2 sin(2alfa).