Question

3 sin 2х + cos 2х = cos^2 x;

Answer 1

Ответ:

Пошаговое объяснение: сократим равные челены в обеих частях

3sin2x=0

Разделите обе части на 3

Sin2x=0

Т.к sin(t)=0 для t=kn (n это число пи) k€Z то 2x=kn

2x=kn, k€Z

Разделим обе стороны на 2

x= kn/2, k€Z это и есть ответ

Answer 2

To solve the trigonometric equation 3sin(2x) + cos(2x) = cos^2(x), we can use trigonometric identities and algebraic manipulations to simplify and solve for x.

Let's start by applying the double angle identity for sine and cosine:

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

Substitute these identities into the equation:

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

Now, simplify the equation:

6sin(x)cos(x) + cos^2(x) - sin^2(x) = cos^2(x)

Since cos^2(x) + sin^2(x) = 1, we can substitute sin^2(x) with 1 - cos^2(x):

6sin(x)cos(x) + cos^2(x) - (1 - cos^2(x)) = cos^2(x)

Simplify further:

6sin(x)cos(x) + 2cos^2(x) - 1 = cos^2(x)

Now, isolate the terms involving cos(x):

6sin(x)cos(x) + 2cos^2(x) - cos^2(x) - 1 = 0

Combine the like terms involving cos(x):

2cos^2(x) + 5sin(x)cos(x) - 1 = 0

We now have a quadratic-like equation in terms of cos(x) and sin(x). Unfortunately, this equation does not have a simple algebraic solution for x. It might be best to use numerical methods or a graphing calculator to approximate the solutions.

If you have specific values or a range for x that you are interested in, we can proceed further using numerical methods or provide an approximation.