Question

Cos ²x - 0,5 sin x -0,5 = 0

Answer 1

Вроде так.....................

Answer 2

To solve the equation cos²x - 0.5sinx - 0.5 = 0, we can use some trigonometric identities and algebraic manipulation.

Let's make a substitution: Let y = sin(x). Then, we can rewrite the equation as follows:

cos²x - 0.5sinx - 0.5 = 0 cos²x - 0.5y - 0.5 = 0

Now, we can use the identity: cos²x + sin²x = 1. Rearrange it to express cos²x in terms of sinx:

cos²x = 1 - sin²x

Substitute this into the equation:

(1 - sin²x) - 0.5y - 0.5 = 0

Now, we have a quadratic equation in terms of y (sinx):

1 - sin²x - 0.5y - 0.5 = 0

Rearrange it to get:

sin²x + 0.5y = 0.5

Next, let's use the double angle formula for sine:

sin(2x) = 2sin(x)cos(x)

Substitute y = sin(x):

sin(2x) = 2y*cos(x)

Now, square both sides of the equation:

sin²(2x) = 4y²*cos²(x)

Since we have a relation between sin²x and y, we can express sin²x in terms of y:

sin²(2x) = 4y²*(1 - sin²x)

Now, substitute sin²x + 0.5y = 0.5:

sin²(2x) = 4y²*(1 - (0.5 - 0.5y))

Simplify further:

sin²(2x) = 4y²*(0.5 + 0.5y)

sin²(2x) = 2y² + 2y³

Now, recall the double angle formula for sine:

sin(2x) = 2sin(x)cos(x)

Using y = sin(x):

sin(2x) = 2y*cos(x)

Now, we have a relation between sin(2x) and y. Let's substitute this into the equation:

(2y*cos(x))² = 2y² + 2y³

4y²*cos²(x) = 2y² + 2y³

Now, divide both sides by 2y²:

2*cos²(x) = 1 + y

Next, recall the Pythagorean identity:

sin²(x) + cos²(x) = 1

Rearrange it to express cos²(x) in terms of sin²(x):

cos²(x) = 1 - sin²(x)

Substitute this into the equation:

2*(1 - sin²(x)) = 1 + y

Now, isolate y:

y = 2 - 2sin²(x)

Now, we have an expression for y in terms of sin(x). Let's go back to our initial substitution y = sin(x):

sin(x) = 2 - 2sin²(x)

This is a cubic equation in terms of sin(x). To solve it, we'll need to apply cubic solving methods. However, the solution process is quite involved, and the equation does not have a simple analytical solution. It is likely that you would need to use numerical methods to find approximate solutions for x.

If you have specific values or constraints for the equation, we can proceed further to find numerical solutions or approximate solutions using numerical methods like Newton's method or other root-finding algorithms.