Question

Sin² x + 4 cos x = 4

Answer 1

Ответ:

Решите триг. уравнение.

Answer 2

4 cos x = 4 - sin²x 

4cosx + sin²x - 4 =0

4cosx + (1 - cos²x) - 4 =0

-cos²x+4cosx - 3 =0 |*(-1)

cos²x - 4cosx + 3 =0

Замена: cosx=t

t² - 4t + 3 =0

D= 16 - 4*1*3 = 16 - 12 = 4

t₁ = 4+2/2 = 3

t₂= 4-2/2 = 1

cosx=3 - нет решения,т.к  -1≤cosx≤1 

cosx= 1

x= 2πn, n∈z

Ответ: 2πn, n∈z.

Answer 3

To solve the equation sin² x + 4 cos x = 4, we can use the trigonometric identity sin² x + cos² x = 1. Let's manipulate the given equation to make use of this identity.

Starting with the given equation: sin² x + 4 cos x = 4

Rearranging terms, we have: sin² x = 4 - 4 cos x

Now, using the identity sin² x + cos² x = 1, we can substitute: 1 - cos² x = 4 - 4 cos x

Rearranging terms again: cos² x - 4 cos x + 3 = 0

Factoring the quadratic equation, we get: (cos x - 3)(cos x - 1) = 0

Now we have two possibilities for solving the equation:

1. cos x - 3 = 0: cos x = 3 However, the range of cosine function lies between -1 and 1, so there are no solutions for this case.

2. cos x - 1 = 0: cos x = 1 This equation has a solution: x = 0.

Therefore, the equation sin² x + 4 cos x = 4 has one solution: x = 0.