Question

2cos в квадрате x - 5 cosx+2=0 ребят помогите решить!!)))

Answer 1

а решается объяснить?
cos (x) = 1/2 
x= pi/3 + 2pi*k/ kпрринадлежит множеству целых чисел

Answer 2

Solving the Equation 2cos^2(x) - 5cos(x) + 2 = 0

To solve the equation 2cos^2(x) - 5cos(x) + 2 = 0, we can use various methods such as factoring, quadratic formula, or trigonometric identities. Let's explore these methods:

Method 1: Factoring

To factor the equation, we need to find two values that multiply to give 2 and add up to -5. However, factoring may not be the most straightforward method for this equation.

Method 2: Quadratic Formula

The quadratic formula can be used to solve any quadratic equation of the form ax^2 + bx + c = 0, where a, b, and c are constants. The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

For the equation 2cos^2(x) - 5cos(x) + 2 = 0, we can identify a = 2, b = -5, and c = 2. Plugging these values into the quadratic formula, we get:

x = (-(-5) ± √((-5)^2 - 4(2)(2))) / (2(2))

Simplifying further:

x = (5 ± √(25 - 16)) / 4

x = (5 ± √9) / 4

x = (5 ± 3) / 4

This gives us two possible solutions:

x = (5 + 3) / 4 = 8 / 4 = 2

x = (5 - 3) / 4 = 2 / 4 = 0.5

Therefore, the solutions to the equation 2cos^2(x) - 5cos(x) + 2 = 0 are x = 2 and x = 0.5.

Method 3: Trigonometric Identities

We can also use trigonometric identities to solve the equation. By rewriting cos^2(x) in terms of 1 - sin^2(x), we can transform the equation into a quadratic equation in terms of sin(x). Let's see how:

2cos^2(x) - 5cos(x) + 2 = 0

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

2(1 - sin^2(x)) - 5cos(x) + 2 = 0

Simplifying further:

2 - 2sin^2(x) - 5cos(x) + 2 = 0

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

Now, we can use the identity sin^2(x) = 1 - cos^2(x) to rewrite the equation:

-2(1 - cos^2(x)) - 5cos(x) + 4 = 0

Simplifying further:

-2 + 2cos^2(x) - 5cos(x) + 4 = 0

2cos^2(x) - 5cos(x) + 2 = 0

We have arrived at the original equation. This means that the solutions obtained using the quadratic formula method are valid.

Therefore, the solutions to the equation 2cos^2(x) - 5cos(x) + 2 = 0 are x = 2 and x = 0.5.

Please note that the solutions may be subject to further verification, and there may be additional solutions depending on the range of x.