Question

Cos2x=2sin² x только без интернета и с объяснением ,пожалуйста

Answer 1

cos2x=cos^2x-sin^2x

Перенесем то что слева вправо, и получим: 

cos^2x-sin^2x-2sin^2x=0

cos^2x-3sin^2x=0

Поделим уравнение на cos^2x

1-3tg^2x=0

tg^2x=1/3

tgx=1/(sqrt3)

x=п/6+пn

tgx=-1/(sqrt3)

x=-п/6+пк

Где sqrt- корень

Answer 2

Solving the Equation cos(2x) = 2sin²(x) + x

To solve the equation cos(2x) = 2sin²(x) + x, we need to find the values of x that satisfy this equation. Let's break down the steps to solve it:

Step 1: Rewrite the equation using trigonometric identities. We can rewrite the equation using the double-angle identity for cosine and the identity for sin²(x):

cos(2x) = 2sin²(x) + x cos²(x) - sin²(x) = 2sin²(x) + x

Step 2: Simplify the equation. Using the Pythagorean identity cos²(x) = 1 - sin²(x), we can substitute it into the equation:

1 - sin²(x) - sin²(x) = 2sin²(x) + x 1 - 2sin²(x) = 3sin²(x) + x

Step 3: Rearrange the equation. To solve for sin²(x), we can move all the terms to one side of the equation:

4sin²(x) + x - 1 = 0

Step 4: Solve the quadratic equation. The equation is now in the form of a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula:

sin²(x) = (-b ± √(b² - 4ac)) / (2a)

In this case, a = 4, b = 1, and c = -1. Substituting these values into the quadratic formula, we get:

sin²(x) = (-1 ± √(1 - 4(4)(-1))) / (2(4)) sin²(x) = (-1 ± √(1 + 16)) / 8 sin²(x) = (-1 ± √17) / 8

Step 5: Solve for sin(x). To find the values of sin(x), we take the square root of both sides:

sin(x) = ±√((-1 ± √17) / 8)

Step 6: Solve for x. Finally, to find the values of x, we can use the inverse sine function (sin⁻¹) to solve for x:

x = sin⁻¹(±√((-1 ± √17) / 8))

Please note that the solutions for x will depend on the range of x you are considering. The inverse sine function has multiple solutions within a given range.

I hope this explanation helps! Let me know if you have any further questions.