Question

√3sin2x+sin²x=cos²x;

Answer 1

Всё решение - на фотографии

Answer 2

To solve the equation √3sin(2x) + sin²(x) = cos²(x), we can use trigonometric identities and algebraic manipulation. Here are the steps to solve this equation:

1. Start with the given equation: √3sin(2x) + sin²(x) = cos²(x)

2. Use the double angle identity for sine: sin(2x) = 2sin(x)cos(x)

3. Replace sin(2x) with 2sin(x)cos(x) in the equation: √3(2sin(x)cos(x)) + sin²(x) = cos²(x)

4. Expand and simplify the equation: 2√3sin(x)cos(x) + sin²(x) = cos²(x)

5. Now, use the Pythagorean identity for sine: sin²(x) + cos²(x) = 1

   So, sin²(x) = 1 - cos²(x)

6. Substitute sin²(x) in terms of cos²(x) in the equation: 2√3sin(x)cos(x) + (1 - cos²(x)) = cos²(x)

7. Move all terms to one side of the equation: 2√3sin(x)cos(x) + 1 - cos²(x) - cos²(x) = 0

8. Combine like terms: 2√3sin(x)cos(x) - 2cos²(x) + 1 = 0

9. Now, factor out common terms: 2(cos(x))(√3sin(x) - cos(x)) + 1 = 0

10. Now, set each factor equal to zero and solve for x:

First factor: 2(cos(x)) = 0 cos(x) = 0

Second factor: √3sin(x) - cos(x) = 0 √3sin(x) = cos(x)

11. Solve for cos(x) = 0: x = π/2 + πn, where n is an integer.

12. Solve for √3sin(x) = cos(x): sin(x)/cos(x) = 1/√3 tan(x) = 1/√3

x = π/6 + πn, where n is an integer.

So, the solutions to the equation √3sin(2x) + sin²(x) = cos²(x) are:

x = π/2 + πn, where n is an integer, and x = π/6 + πn, where n is an integer.