Question

Cos2x+sin x= 1+sin x *cos x

Answer 1

Ответ:

решение представлено на фото

Answer 2

To solve the equation cos(2x) + sin(x) = 1 + sin(x) * cos(x), we can simplify it step by step.

Let's start by using the double-angle identity for cosine, which states that cos(2x) = 1 - 2sin^2(x). Replacing cos(2x) in the equation, we get:

1 - 2sin^2(x) + sin(x) = 1 + sin(x) * cos(x)

Rearranging the terms, we have:

-2sin^2(x) + sin(x) * cos(x) - sin(x) = 0

Factoring out sin(x) from the left side:

sin(x) * (-2sin(x) + cos(x) - 1) = 0

Now, there are two possibilities for the equation to be true:

1. sin(x) = 0
2. -2sin(x) + cos(x) - 1 = 0

For the first possibility, sin(x) = 0, the solutions are x = 0 and x = π (plus any integer multiple of 2π).

For the second possibility, -2sin(x) + cos(x) - 1 = 0, we need to solve this trigonometric equation.

Rearranging the equation, we have:

cos(x) = 2sin(x) - 1

Using the identity sin^2(x) + cos^2(x) = 1, we can substitute cos^2(x) with (1 - sin^2(x)):

1 - sin^2(x) = (2sin(x) - 1)^2

Expanding and simplifying:

1 - sin^2(x) = 4sin^2(x) - 4sin(x) + 1

Rearranging the terms:

5sin^2(x) - 4sin(x) = 0

Now, we can factor out sin(x):

sin(x)(5sin(x) - 4) = 0

So we have two possibilities:

1. sin(x) = 0
2. 5sin(x) - 4 = 0

For the first possibility, sin(x) = 0, we have already included the solutions x = 0 and x = π in the previous step.

For the second possibility, 5sin(x) - 4 = 0, we solve for sin(x):

5sin(x) = 4 sin(x) = 4/5

Taking the inverse sine (sin^-1) of both sides, we find:

x = sin^-1(4/5)

Therefore, the complete set of solutions for the given equation is:

x = 0, π, sin^-1(4/5) (plus any integer multiple of 2π).