Question

9sinx * cos x - 7 cos^2x =2 sin^2 x Решите

Answer 1

Решение задания смотри на фотографии

Answer 2

Solving the Equation: 9sin(x) * cos(x) - 7cos^2(x) = 2sin^2(x)

To solve the equation 9sin(x) * cos(x) - 7cos^2(x) = 2sin^2(x), we can follow these steps:

Step 1: Simplify the equation using trigonometric identities. Step 2: Rearrange the equation to isolate the trigonometric function. Step 3: Solve for the values of x that satisfy the equation.

Let's go through each step in detail.

Step 1: Simplify the equation using trigonometric identities

We can simplify the equation using the following trigonometric identities:

- sin^2(x) = 1 - cos^2(x) - sin(2x) = 2sin(x)cos(x)

Applying these identities to the equation, we get:

9sin(x) * cos(x) - 7cos^2(x) = 2(1 - cos^2(x))

Simplifying further, we have:

9sin(x) * cos(x) - 7cos^2(x) = 2 - 2cos^2(x)

Step 2: Rearrange the equation to isolate the trigonometric function

To isolate the trigonometric function, we can rearrange the equation as follows:

9sin(x) * cos(x) - 7cos^2(x) - 2 + 2cos^2(x) = 0

Combining like terms, we have:

9sin(x) * cos(x) - 5cos^2(x) - 2 = 0

Step 3: Solve for the values of x that satisfy the equation

To solve the equation, we can factor out common terms:

cos(x)(9sin(x) - 5cos(x)) - 2 = 0

Now, we have two factors:

1. cos(x) = 0 2. 9sin(x) - 5cos(x) = 2

Let's solve each factor separately.

Factor 1: cos(x) = 0

When cos(x) = 0, it means x is equal to π/2 + nπ or 3π/2 + nπ, where n is an integer.

Factor 2: 9sin(x) - 5cos(x) = 2

To solve this equation, we can use the identity sin(2x) = 2sin(x)cos(x). Rearranging the equation, we get:

9sin(x) - 5cos(x) = 2 9sin(x) - 5(2sin(x)cos(x)) = 2 9sin(x) - 10sin(2x) = 2

Now, we can substitute sin(2x) = 2sin(x)cos(x) into the equation:

9sin(x) - 10(2sin(x)cos(x)) = 2 9sin(x) - 20sin(x)cos(x) = 2 sin(x)(9 - 20cos(x)) = 2

We have two factors again:

1. sin(x) = 0 2. 9 - 20cos(x) = 2

Factor 1: sin(x) = 0

When sin(x) = 0, it means x is equal to nπ, where n is an integer.

Factor 2: 9 - 20cos(x) = 2

To solve this equation, we can rearrange it:

20cos(x) = 9 - 2 20cos(x) = 7 cos(x) = 7/20

Using the inverse cosine function, we can find the values of x:

x = arccos(7/20)

Please note that the exact values of x will depend on the range of x you are considering.

In summary, the solutions to the equation 9sin(x) * cos(x) - 7cos^2(x) = 2sin^2(x) are:

1. x = π/2 + nπ or 3π/2 + nπ, where n is an integer. 2. x = arccos(7/20), where x is within the appropriate range.

Please let me know if you need any further clarification or assistance!