Question

2 cos^2 x - 5 cos x=3

Answer 1

Решение во вложении.

Answer 2

To solve the equation 2cos^2(x) - 5cos(x) = 3, we need to find the values of x that satisfy the equation. Here's the step-by-step solution:

Step 1: Rewrite the equation in standard quadratic form by moving all terms to one side:

2cos^2(x) - 5cos(x) - 3 = 0

Step 2: To make it easier to solve, let's introduce a substitution. Let's set y = cos(x):

Now, the equation becomes:

2y^2 - 5y - 3 = 0

Step 3: Solve the quadratic equation for y. To do this, you can factor the quadratic or use the quadratic formula:

Factorization: (2y + 1)(y - 3) = 0

Setting each factor to zero and solving for y:

1. 2y + 1 = 0 => 2y = -1 => y = -1/2
2. y - 3 = 0 => y = 3

Step 4: We found the possible values for y, but we need to find the corresponding values for x. Recall that y = cos(x), so we have two separate equations to solve for x:

1. cos(x) = -1/2
2. cos(x) = 3

Step 5: Solve each equation for x:

1. cos(x) = -1/2: To find the solutions in the interval [0, 2π), we look for angles where the cosine function takes the value -1/2.

These angles are π/3 and 5π/3, as cos(π/3) = cos(5π/3) = -1/2.

2. cos(x) = 3: The value 3 is outside the range of the cosine function, which is always between -1 and 1. Thus, there are no solutions for this equation.

Step 6: Write down the final solutions for x:

x = π/3 + 2πn (where n is an integer) x = 5π/3 + 2πn (where n is an integer)

These are the solutions to the equation 2cos^2(x) - 5cos(x) = 3 in the interval [0, 2π). Keep in mind that cosine has a periodic nature, so you can add multiples of 2π to the solutions to find all possible solutions.