Question

Cos x-cos y=√2__x+y=П.Решыте систему уровнений

Answer 1

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Answer 2

To solve the system of equations:

1. cos(x) - cos(y) = √2
2. x + y = π

We can use the trigonometric identity for the difference of two cosines:

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

Let's use this identity to rewrite equation (1):

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

Now we'll substitute the value of (x + y) from equation (2) into the equation above:

cos(x) - cos(y) = -2 * sin(π / 2) * sin((x - y) / 2) cos(x) - cos(y) = -2 * sin((x - y) / 2)

Now, we can rewrite the system as a set of equations:

1. -2 * sin((x - y) / 2) = √2
2. x + y = π

Now we'll solve the first equation for sin((x - y) / 2):

sin((x - y) / 2) = -√2 / 2

Now, we'll find the angle whose half-angle has a sine of -√2 / 2. The two angles that satisfy this condition are:

1. (x - y) / 2 = -π / 4
2. (x - y) / 2 = 3 * π / 4

Now we can solve for (x - y) in both cases:

1. x - y = -π / 2
2. x - y = 3 * π / 2

Now, we have a system of two equations:

1. x + y = π
2. x - y = -π / 2 or x - y = 3 * π / 2

Let's solve for x and y in both cases:

Case 1: x - y = -π / 2 Adding equations (1) and (2): 2x = π / 2 x = π / 4

Substituting the value of x in equation (1): π / 4 + y = π y = π - π / 4 y = 3π / 4

So, the solutions for Case 1 are x = π / 4 and y = 3π / 4.

Case 2: x - y = 3 * π / 2 Adding equations (1) and (2): 2x = 5 * π / 2 x = 5 * π / 4

Substituting the value of x in equation (1): 5 * π / 4 + y = π y = π - 5 * π / 4 y = -π / 4

So, the solutions for Case 2 are x = 5π / 4 and y = -π / 4.

In conclusion, the system of equations has two sets of solutions:

1. x = π / 4 and y = 3π / 4
2. x = 5π / 4 and y = -π / 4