Question

3cos²x + 5sin²x=8sin x ×cos x

Answer 1

Ответ:

пи/4+пи*к; арктг 3/5 + пи*к

Объяснение:

Добавляю: при cosx=0, sinx=0 уравнение не имеет смысла, так что делим на cos^2x

Answer 2

To solve the equation 3cos²x + 5sin²x = 8sin(x) × cos(x), we can start by applying trigonometric identities to simplify it.

Using the Pythagorean identity sin²x + cos²x = 1, we can rewrite the equation as:

3cos²x + 5(1 - cos²x) = 8sin(x) × cos(x)

Expanding and rearranging the terms:

3cos²x + 5 - 5cos²x = 8sin(x) × cos(x)

Combining like terms:

-2cos²x + 5 = 8sin(x) × cos(x)

Next, we can use the double angle identity for cosine, which states that cos(2x) = 2cos²x - 1. Rearranging this equation, we have:

2cos²x = cos(2x) + 1

Substituting this into our equation:

-(cos(2x) + 1) + 5 = 8sin(x) × cos(x)

Now, let's simplify further:

• cos(2x) - 1 + 5 = 8sin(x) × cos(x)

• cos(2x) + 4 = 8sin(x) × cos(x)

Finally, using the identity for the product of sine and cosine, sin(2x) = 2sin(x) × cos(x), we can rewrite the equation as:

• cos(2x) + 4 = 4sin(2x)

This is a trigonometric equation involving cosine and sine functions. To solve for x, we'll need to use trigonometric properties and techniques such as the sum-to-product or product-to-sum identities, as well as factoring, substitution, or numerical approximation methods. The exact solution or further simplification depends on the specific form and constraints of the equation.