Question

Sin2x-sin3x+sinx/cos2x-cos3x+cos4x решите, пожалуйста

Answer 1

$\frac{Sin2x-Sin3x-Sinx}{Cos2x-Cos3x+Cos4x} = \frac{Sin2x -(Sin3x +Sinx)}{(Cos2x+Cos4x)-Cos3x}= \frac{Sin2x-2Sin \frac{3x+x}{2}Cos \frac{3x-x}{2} }{2Cos \frac{2x+4x}{2}Cos \frac{2x-4x}{2} -Cos3x }$$= \frac{Sin2x-2Sin2xCosx}{2Cos3xCosx-Cos3x} = \frac{Sin2x(1-2Cosx)}{Cos3x(2Cosx-1)}=- \frac{Sin2x}{Cos3x}$

Answer 2

Problem Analysis

To solve the equation sin(2x) - sin(3x) + sin(x) / cos(2x) - cos(3x) + cos(4x), we need to simplify the expression and find the values of x that satisfy the equation.

Simplifying the Expression

Let's simplify the expression step by step:

1. Combine the terms with the same trigonometric functions: - sin(2x) - sin(3x) + sin(x) / cos(2x) - cos(3x) + cos(4x) - Rearrange the terms: (sin(2x) + sin(x)) / (cos(2x) - cos(3x) + cos(4x)) - sin(3x) - Apply the sum-to-product trigonometric identities: 2sin(x)cos(x) + sin(x) / 2cos^2(x) - 1 + cos^2(x) - sin^2(x) - Simplify further: 2sin(x)cos(x) + sin(x) / 2cos^2(x) - sin^2(x) - Apply the Pythagorean identity: 2sin(x)cos(x) + sin(x) / 2(1 - sin^2(x)) - sin^2(x) - Simplify: 2sin(x)cos(x) + sin(x) / 2 - sin^3(x) / 2 - sin^2(x)

2. Combine like terms: 2sin(x)cos(x) + sin(x) - sin^3(x) / 2 - sin^2(x) / 2

3. Factor out sin(x): sin(x)(2cos(x) + 1 - sin^2(x) / 2 - sin(x) / 2)

4. Apply the Pythagorean identity again: sin(x)(2cos(x) + 1 - (1 - cos^2(x)) / 2 - sin(x) / 2) sin(x)(2cos(x) + 1 - 1 + cos^2(x) / 2 - sin(x) / 2) sin(x)(cos^2(x) + cos(x) - sin(x) / 2)

5. Simplify further: sin(x)(cos^2(x) + cos(x) - sin(x) / 2)

Solving the Equation

To solve the equation, we need to find the values of x that make the expression equal to zero. Therefore, we set the expression equal to zero and solve for x:

sin(x)(cos^2(x) + cos(x) - sin(x) / 2) = 0

To find the values of x, we set each factor equal to zero and solve for x:

1. sin(x) = 0 - This occurs when x = 0, π, 2π, 3π, ...

2. cos^2(x) + cos(x) - sin(x) / 2 = 0 - This equation does not have a simple algebraic solution. We can use numerical methods or graphing to find approximate solutions.

Conclusion

The equation sin(2x) - sin(3x) + sin(x) / cos(2x) - cos(3x) + cos(4x) simplifies to sin(x)(cos^2(x) + cos(x) - sin(x) / 2). The values of x that satisfy the equation are x = 0, π, 2π, 3π, ... and the solutions to the equation cos^2(x) + cos(x) - sin(x) / 2 = 0 can be found using numerical methods or graphing.