Question

Sin x *cos x=1/2sin x

Answer 1

$\sin{x} \cos{x} = \frac{1}{2} \sin{x} \\ \sin{x}( \cos{x} - \frac{1}{2} ) = 0$
Произведение равно, если хотя бы один из множителей равен нулю, а другой при этом не теряет смысл. Получаем,
sinx=0 или cosx-(1/2)=0
1) sinx=0
x=Πn, n-целое число
2) сosx-(1/2)=0
cosx=1/2
x =-+(Π/3)+2Πk, k - целое число.
Ответ: Πn, n -целое; -+(Π/3)+2Πk, k-целое.

Answer 2

To solve the equation sin(x) * cos(x) = (1/2)sin(x), we can manipulate the equation algebraically.

Let's start by dividing both sides of the equation by sin(x):

sin(x) * cos(x) / sin(x) = (1/2)sin(x) / sin(x)

cos(x) = 1/2

Now, we can solve for x by taking the inverse cosine (arccos) of both sides:

x = arccos(1/2)

The value of arccos(1/2) is π/3 (or 60 degrees) because the cosine function has a value of 1/2 at π/3 and 5π/3.

Therefore, the solution to the equation sin(x) * cos(x) = (1/2)sin(x) is:

x = π/3 + 2πn

where n is an integer representing any additional solutions.