Question

2 cos^2 x = 1 + sin x

Answer 1

Ответ:

Решение тригонометрических уравнений.

Answer 2

To solve the trigonometric equation 2 cos^2(x) = 1 + sin(x), we'll need to manipulate the equation to find the values of x that satisfy it. Here's how you can do it step by step:

Step 1: Use the Pythagorean identity for cos^2(x):

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

Step 2: Substitute sin^2(x) with (1 - cos^2(x)) in the original equation:

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

Step 3: Now, simplify the equation:

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

Step 4: Move all the terms to one side of the equation:

2 cos^2(x) + cos^2(x) = 2

Step 5: Combine like terms:

3 cos^2(x) = 2

Step 6: Divide both sides by 3:

cos^2(x) = 2/3

Step 7: Take the square root of both sides:

cos(x) = ±√(2/3)

Step 8: Solve for x:

x = cos^(-1)(√(2/3)) or x = cos^(-1)(-√(2/3))

Note: cos^(-1) denotes the inverse cosine function, also known as arccosine, and it gives us the angles whose cosine is the given value.

Now, to find the numerical values for x, use a calculator:

x ≈ 0.8411 rad or x ≈ -0.8411 rad (approximately)

Depending on the context or the range you are interested in, the solutions can be further expanded, considering the periodicity of the trigonometric functions. For example, if you want all solutions in the interval [0, 2π), you can use the unit circle and the symmetry of the cosine function to find other solutions.