Question

Sin^2 x + 4cos x=2,75​

Answer 1

Ответ:

$x = \frac{\pi}{3} + 2k\pi$

Answer 2

To solve the equation sin^2(x) + 4cos(x) = 2.75, you can use trigonometric identities to simplify it.

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

Now, we need to rewrite the equation in terms of either sin(x) or cos(x) so that we can apply this identity. Let's rewrite cos(x) in terms of sin(x) using the identity: cos^2(x) = 1 - sin^2(x).

So, we have:

sin^2(x) + 4(1 - sin^2(x)) = 2.75

Now, distribute the 4 on the left side of the equation:

sin^2(x) + 4 - 4sin^2(x) = 2.75

Combine like terms:

-3sin^2(x) + 4 = 2.75

Subtract 4 from both sides:

-3sin^2(x) = 2.75 - 4

-3sin^2(x) = -1.25

Now, divide by -3:

sin^2(x) = (-1.25) / (-3)

sin^2(x) = 1.25 / 3

sin^2(x) = 5/12

To solve for sin(x), take the square root of both sides:

sin(x) = ±√(5/12)

Now, we need to determine the sign. Since sin(x) is positive in the first and second quadrants, you have:

sin(x) = √(5/12) or sin(x) = -√(5/12)

Now, to find the values of x, you can take the arcsin (inverse sine) of these values:

For sin(x) = √(5/12): x = arcsin(√(5/12))

For sin(x) = -√(5/12): x = arcsin(-√(5/12))

These values of x will give you the solutions to the equation sin^2(x) + 4cos(x) = 2.75. Remember to check for any restrictions on the domain of x, typically between -π and π or 0 and 2π depending on the context of the problem.