Question

Cos x + cos 2y = -0,5,3 cos 2y - cos x = 2,5;​

Answer 1

Ответ:    x = π + 2π n ,   y = ± π/6 + π n ,    nЄ Z .

Пошаговое объяснение:

{ cos x + cos 2y = -0,5,

{ 3 cos 2y - cos x = 2,5;​

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4cos2y = 2 ;

cos2y  = 1/2 ;

2y = ± π/3 + 2π n ,    nЄ Z

y = ± π/6 + π n ,    nЄ Z ;

Перше рівняння даної системи помножимо на ( - 3 ) і додамо до 2- го :

- 4cosx = 4 ;

cosx = - 1 ;

x = π + 2π n ,    nЄ Z .

Answer 2

To solve the system of equations:

1. cos(x) + cos(2y) = -0.5
2. 3cos(2y) - cos(x) = 2.5

You can use the method of substitution or elimination. Let's use the elimination method. We can add the two equations together to eliminate the 'cos(x)' term:

(cos(x) + cos(2y)) + (3cos(2y) - cos(x)) = (-0.5) + 2.5

This simplifies to:

4cos(2y) = 2

Now, divide both sides by 4 to solve for cos(2y):

cos(2y) = 2/4 cos(2y) = 0.5

Now, you need to find the values of '2y' that satisfy this equation. The cosine of an angle is positive in the first and fourth quadrants, and it's equal to 0.5 at 60 degrees (π/3 radians) and 300 degrees (5π/3 radians) in those quadrants.

So, we have two possible values for 2y:

1. 2y = π/3
2. 2y = 5π/3

Now, divide each equation by 2 to find the values of 'y':

1. y = π/6
2. y = 5π/6

Now that we have the values for 'y', we can find the corresponding values for 'x' using the original equations. Let's use the first equation:

1. cos(x) + cos(2y) = -0.5

For y = π/6:

cos(x) + cos(2 * (π/6)) = -0.5 cos(x) + cos(π/3) = -0.5

For y = 5π/6:

cos(x) + cos(2 * (5π/6)) = -0.5 cos(x) + cos(5π/3) = -0.5

Now, find the values of 'x' for each case:

1. cos(x) + cos(π/3) = -0.5 cos(x) = -0.5 - cos(π/3)

Use the values of 'x' for which cos(x) is equal to this expression.

2. cos(x) + cos(5π/3) = -0.5 cos(x) = -0.5 - cos(5π/3)

Use the values of 'x' for which cos(x) is equal to this expression.

Please note that cosine has periodic behavior, so there may be multiple solutions for 'x' in each case. You can use trigonometric properties to find the specific solutions within the desired range.