Question

Тригонометрия 3sin^2 x x-8 sinx cosx +7 cos^2 x =1 решите пожалуйста )

Answer 1

3sin²x-8sinx*cosx+7cos²x=1
3sin²x-8sinx*cosx+7cos²x-sin²x-cos²x=0
2sin²x-8sinx*cosx+6cos²x=0 | :cos²x≠0
2tg²x-8tgx+6=0
tgx=t
2t²-8t+6=0
t²-4t+3=0
t₁=3, t₂=1
1. tgx=3, x₁=arctg3+πn, n∈Z

2. tgx=1, x₂=π/4+πn,n∈Z

Answer 2

Solving the Trigonometric Equation

To solve the equation 3sin^2(x) + x - 8 + sin(x) + cos(x) + 7 + cos^2(x) = 1, we can follow these steps:

Step 1: Simplify the equation by combining like terms and using trigonometric identities if necessary.

Step 2: Rearrange the equation to isolate the trigonometric terms on one side and the constant terms on the other side.

Step 3: Solve the resulting equation using algebraic techniques or numerical methods.

Let's go through these steps in detail.

Step 1: Simplify the Equation

The given equation is: 3sin^2(x) + x - 8 + sin(x) + cos(x) + 7 + cos^2(x) = 1

To simplify the equation, we can combine the trigonometric terms and the constant terms separately.

For the trigonometric terms: - Combine sin^2(x) and cos^2(x) using the identity sin^2(x) + cos^2(x) = 1.

For the constant terms: - Combine -8, 7, and 1 to get 0.

After simplifying, the equation becomes: 3sin^2(x) + cos^2(x) + sin(x) + cos(x) + x - 8 = 0

Step 2: Rearrange the Equation

Now, let's rearrange the equation to isolate the trigonometric terms on one side and the constant terms on the other side.

3sin^2(x) + cos^2(x) + sin(x) + cos(x) + x - 8 = 0

Rearranging the terms, we get: 3sin^2(x) + sin(x) + cos^2(x) + cos(x) + x - 8 = 0

Step 3: Solve the Equation

To solve the equation, we can use numerical methods or algebraic techniques. However, since the equation is quite complex, it may not have a simple algebraic solution.

One approach is to use numerical methods, such as graphing or using a calculator or software like MATLAB. These methods can help us find approximate solutions.

Another approach is to use trigonometric identities and algebraic techniques to simplify the equation further. However, in this case, the equation is already simplified as much as possible.

Unfortunately, without additional information or constraints, it is not possible to provide an exact solution to the equation.

If you have access to MATLAB, you can use it to solve the equation numerically. MATLAB provides various functions and tools for solving equations, including the `solve` function. You can input the equation into MATLAB and use the `solve` function to find the solutions.

Here's an example of how you can use MATLAB to solve the equation:

syms x
eqn = 3*sin(x)^2 + sin(x) + cos(x)^2 + cos(x) + x - 8 == 0;
sol = solve(eqn, x);
sol

This code defines a symbolic variable `x`, sets up the equation using the symbolic variable, and then uses the `solve` function to find the solutions. The solutions will be displayed in the `sol` variable.

Please note that the exact solutions to the equation may be complex and involve transcendental functions.