Question

Решить систему: { x+y = 2π/3 { cos(x+y) + cos(x-y) = 1/2

Answer 1

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Answer 2

To solve the system of equations:

1. x + y = 2π/3
2. cos(x + y) + cos(x - y) = 1/2

We can use trigonometric identities to simplify the second equation and then solve the system.

Step 1: Use the sum-to-product trigonometric identity for the second equation.

cos(A) + cos(B) = 2 * cos((A + B)/2) * cos((A - B)/2)

Applying this identity to equation (2):

cos(x + y) + cos(x - y) = 2 * cos((x + y + x - y)/2) * cos((x + y - x + y)/2) cos(x + y) + cos(x - y) = 2 * cos(x) * cos(y)

So now we have:

2 * cos(x) * cos(y) = 1/2

Step 2: Rewrite the first equation to solve for y:

y = 2π/3 - x

Step 3: Substitute the value of y from the first equation into the second equation:

2 * cos(x) * cos(2π/3 - x) = 1/2

Step 4: Use the cosine double-angle formula:

cos(2θ) = 2 * cos^2(θ) - 1

So, cos(2π/3 - x) = 2 * cos^2((2π/3 - x)/2) - 1

Step 5: Simplify:

cos(2π/3 - x) = 2 * cos^2(π/3 - x/2) - 1

Step 6: Substitute this back into the equation:

2 * cos(x) * [2 * cos^2(π/3 - x/2) - 1] = 1/2

Step 7: Now, solve for cos(x):

4 * cos(x) * cos^2(π/3 - x/2) - 2 * cos(x) = 1/2 4 * cos(x) * cos^2(π/3 - x/2) - 2 * cos(x) - 1/2 = 0

Step 8: Let's make a substitution to simplify the equation further:

Let a = π/3 - x/2

Then, cos(x) = cos(π/3 - 2a)

Step 9: Rewrite the equation in terms of 'a':

4 * cos(π/3 - 2a) * cos^2(a) - 2 * cos(π/3 - 2a) - 1/2 = 0

Step 10: Multiply everything by 2 to clear the fraction:

8 * cos(π/3 - 2a) * cos^2(a) - 4 * cos(π/3 - 2a) - 1 = 0

Step 11: Use the double-angle formula for cosine:

cos(π - θ) = -cos(θ)

So, cos(π/3 - 2a) = -cos(2a - π/3)

Step 12: Substitute this back into the equation:

8 * (-cos(2a - π/3)) * cos^2(a) - 4 * (-cos(2a - π/3)) - 1 = 0

Step 13: Distribute and simplify:

-8 * cos(2a - π/3) * cos^2(a) + 4 * cos(2a - π/3) - 1 = 0

Step 14: Rearrange the terms:

8 * cos(2a - π/3) * cos^2(a) - 4 * cos(2a - π/3) + 1 = 0

Step 15: Factor the equation:

(2 * cos(2a - π/3) - 1)^2 = 0

Step 16: Take the square root:

2 * cos(2a - π/3) - 1 = 0

Step 17: Solve for cos(2a - π/3):

2 * cos(2a - π/3) = 1

cos(2a - π/3) = 1/2

Step 18: Now find the value of (2a - π/3):

2a - π/3 = ±π/3 + 2nπ (where n is an integer)

Step 19: Solve for 'a':

2a = 2π/3 + π/3 + 2nπ 2a = π + 2nπ a = π/2 + nπ

Step 20: Find the corresponding values of 'x':

π/3 - x/2 = π/2 + nπ x/2 = -π/6 + nπ x = -π/3 + 2nπ

Step 21: Now, find the corresponding values of 'y':

y = 2π/3 - x y = 2π/3 - (-π/3 + 2nπ) y = 3π/3 + π/3 - 2nπ y = 4π/3 - 2nπ

So, the solutions to the system of equations are:

1. x = -π/3 + 2nπ, y = 4π/3 - 2nπ
2. x = -π/3 + 2nπ, y = 4π/3 - 2nπ, where n is an integer.