Question

Cos3x-cos7x=корень из 3 синусов2х Даю 45 баллов, помогите

Answer 1

Воспользуемся формулой "разность косинусов = удвоенное произведение синуса полусуммы на синус обратной полуразности":

2sin 5x sin 2x=√3 sin 2x;
sin 2x (2sin 5x-√3)=0;
sin 2x=0⇒2x=πn⇒x=πn/2
или 2sin 5x - √3=0⇒sin 5x=√3/2⇒5x=π/3+2πk или 5x=2π/3+2πm; то есть
x=π/15+2πk/5 или x=2π/15+2πm/5

Ответ: πn/2, n∈Z ; x=π/15+2πk/5, k∈Z; 2π/15+2πm/5, m∈Z

Answer 2

Solving the Equation Cos(3x) - Cos(7x) = √3sin(2x)

To solve the equation Cos(3x) - Cos(7x) = √3sin(2x), we will use trigonometric identities and algebraic manipulation. Let's break down the steps:

Step 1: Convert the equation to a single trigonometric function. Step 2: Simplify the equation using trigonometric identities. Step 3: Solve for x.

Step 1: Convert the Equation to a Single Trigonometric Function

To convert the equation to a single trigonometric function, we can use the following trigonometric identity:

Cos(A) - Cos(B) = -2sin((A + B)/2)sin((A - B)/2)

Applying this identity to our equation, we get:

-2sin((3x + 7x)/2)sin((3x - 7x)/2) = √3sin(2x)

Simplifying further, we have:

-2sin(5x/2)sin(-2x/2) = √3sin(2x)

Since sin(-θ) = -sin(θ), we can simplify the equation to:

2sin(5x/2)sin(x) = √3sin(2x)

Step 2: Simplify the Equation Using Trigonometric Identities

To simplify the equation further, we can use the following trigonometric identity:

sin(2θ) = 2sin(θ)cos(θ)

Applying this identity to our equation, we get:

2sin(5x/2)sin(x) = √3(2sin(x)cos(x))

Simplifying further, we have:

2sin(5x/2)sin(x) = 2√3sin(x)cos(x)

Dividing both sides of the equation by 2sin(x), we get:

sin(5x/2) = √3cos(x)

Step 3: Solve for x

To solve for x, we need to find the values of x that satisfy the equation sin(5x/2) = √3cos(x).

Unfortunately, without additional information or constraints, it is not possible to determine the exact values of x that satisfy the equation. The equation represents an infinite number of solutions.

To find specific solutions, we would need additional information, such as a range for x or specific conditions.

If you have any additional information or constraints, please provide them, and I will be happy to assist you further.