Question

2+cosквадратx=2sinx​

Answer 1

Ответ:

x=П/2+2Пk

Объяснение:

косинус расписывается как 1+sin(x)^2

при упрощении получаем sin(x)^2+2sin(x)-3=0

введём замену, получим t=sin(x)     t^2+2t-3=0

$\left \{ {{t=-3} \atop {t=1}} \right.$

получаем:

sin(x)=-3

sin(x)=1

отсюда:

x не принадлежит рациональному числу

x=П/2+2Пk

Answer 2

To solve the equation 2 + cos^2(x) = 2sin(x), we can use trigonometric identities to simplify it. Let's work through the steps:

First, recall the Pythagorean identity: sin^2(x) + cos^2(x) = 1.

Rearranging this equation, we get: cos^2(x) = 1 - sin^2(x).

Now substitute this expression for cos^2(x) in the original equation:

2 + (1 - sin^2(x)) = 2sin(x).

Simplifying further:

3 - sin^2(x) = 2sin(x).

Rearrange the equation by moving all terms to one side:

sin^2(x) + 2sin(x) - 3 = 0.

Now, let's substitute u = sin(x):

u^2 + 2u - 3 = 0.

Factorize the quadratic equation:

(u + 3)(u - 1) = 0.

Setting each factor equal to zero:

u + 3 = 0 or u - 1 = 0.

Solving for u in each case:

u = -3 or u = 1.

Now, substitute back u = sin(x):

sin(x) = -3 or sin(x) = 1.

Since the range of sine function is [-1, 1], we can conclude that sin(x) = 1 is the only valid solution.

Therefore, the solution to the equation 2 + cos^2(x) = 2sin(x) is:

sin(x) = 1.