Question

Решение простейших тригонометрических уравнений 2cosх-3sinx*cosx=0 ,помогите(

Answer 1

cosx=0 x=П/2(2k+1)

2=3sinx

sinx=2/3

x=arcsin(2/3)+2pik

x=П-arcsin(2/3)+2pik

Answer 2

Solution to the Trigonometric Equation 2cos(x) - 3sin(x)cos(x) = 0

To solve the trigonometric equation 2cos(x) - 3sin(x)cos(x) = 0, we can use trigonometric identities and algebraic manipulations. Let's break down the steps to find the solution:

Step 1: Factor out the common term cos(x) from the equation: cos(x)(2 - 3sin(x)) = 0.

Step 2: Set each factor equal to zero and solve for x: - cos(x) = 0 (Equation 1) - 2 - 3sin(x) = 0 (Equation 2)

Now, let's solve each equation separately:

# Solving Equation 1: cos(x) = 0

The solutions to this equation occur when the cosine function equals zero. From trigonometric properties, we know that the cosine function equals zero at specific angles. The general solution for this equation is:

x = (2n + 1) * π/2, where n is an integer.

This means that x can take on values such as π/2, 3π/2, 5π/2, and so on.

# Solving Equation 2: 2 - 3sin(x) = 0

To solve this equation, we need to isolate the sin(x) term. Let's rearrange the equation:

3sin(x) = 2.

Divide both sides of the equation by 3:

sin(x) = 2/3.

The solutions to this equation occur when the sine function equals 2/3. From trigonometric properties, we know that the sine function equals 2/3 at specific angles. The general solution for this equation is:

x = arcsin(2/3) + 2πn or x = π - arcsin(2/3) + 2πn, where n is an integer.

This means that x can take on values such as arcsin(2/3), π - arcsin(2/3), arcsin(2/3) + 2π, π - arcsin(2/3) + 2π, and so on.

Summary of Solutions

The solutions to the trigonometric equation 2cos(x) - 3sin(x)cos(x) = 0 are given by the combination of the solutions to Equation 1 and Equation 2:

- x = (2n + 1) * π/2, where n is an integer (from Equation 1). - x = arcsin(2/3) + 2πn or x = π - arcsin(2/3) + 2πn, where n is an integer (from Equation 2).

Please note that these solutions are general solutions and represent an infinite number of possible values for x. To find specific solutions within a given range, you may need to further restrict the values of n or consider the domain of the problem.

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