Question

Cos2x * sin2x - 1 = -1/2

Answer 1

Решение задания смотри на фотографии

Answer 2

To solve the equation cos(2x) * sin(2x) - 1 = -1/2, we can use trigonometric identities and algebraic manipulations. Let's break down the steps:

Step 1: Simplify the equation using trigonometric identities. The identity sin(2x) = 2sin(x)cos(x) can be used to rewrite the equation as follows:

cos(2x) * 2sin(x)cos(x) - 1 = -1/2

Step 2: Expand and rearrange the equation. By distributing cos(2x) to both terms, we get:

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

Step 3: Substitute the identity cos^2(x) = 1 - sin^2(x). Using this identity, we can rewrite the equation further:

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

Step 4: Simplify the equation. Expanding the equation, we have:

2sin(x) - 2sin^3(x) - 1 = -1/2

Multiplying everything by 2 to eliminate the fractions gives:

4sin(x) - 4sin^3(x) - 2 = -1

Rearranging the terms, we obtain:

4sin^3(x) - 4sin(x) + 1 = 0

Step 5: Solve the cubic equation. At this point, we have a cubic equation in terms of sin(x). Solving this equation analytically can be challenging. However, we can use numerical methods or approximation techniques to find the solutions.

Alternatively, we can use a graphing calculator or software to plot the equation and find the approximate solutions.

Keep in mind that the solutions to this equation might involve multiple values of x, depending on the range you are considering.

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