Question

Sinx - siny = 0.5 cosx + cosy = √(3)/2

Answer 1

Ответ:

(π/2 + 2πm; π/6 + 2πn)

(-π/6 + 2πm; -π/2 + 2πn); n,m∈Z

Объяснение:Применяем к 1-му уравнение "разность синусов", а ко 2-му "сумму косинусов":

(1)  

(2)  

Делим почленно (1) на (2):

(3)

(x - y)/2 = π/6 + πk, k∈Z

x = y + π/3 + 2πk - Подставляем в (1):

2·sin(0.5·(y + π/3 + 2πk - y)·cos(0.5·(y + π/3 + 2πk + y)) = 1/2

2·sin(π/6)·cos(y + π/6) = 1/2

cos(y + π/6) = 1/2

y + π/6 = ±π/3 + 2πn, n∈Z

1) y = -π/2 + 2πn

x = -π/6 + 2πn + 2πk = -π/6 + 2πm, m∈Z

или

2) y = π/6 + 2πn

x = π/2 + 2πn + 2πk = π/2 + 2πm

Проверяем получившиеся корни - все подходят

Answer 2

To solve the equations sin(x) - sin(y) = 0.5 and cos(x) + cos(y) = √(3)/2, we can use trigonometric identities and algebraic manipulation.

Let's start with the first equation: sin(x) - sin(y) = 0.5.

Using the trigonometric identity for the difference of two sines, we have:

2 sin((x-y)/2) cos((x+y)/2) = 0.5.

Dividing both sides by 2, we get:

sin((x-y)/2) cos((x+y)/2) = 0.25.

Now, let's move on to the second equation: cos(x) + cos(y) = √(3)/2.

Using the trigonometric identity for the sum of two cosines, we have:

2 cos((x+y)/2) cos((x-y)/2) = √(3)/2.

Dividing both sides by 2, we get:

cos((x+y)/2) cos((x-y)/2) = √(3)/4.

Now, we have two equations:

1. sin((x-y)/2) cos((x+y)/2) = 0.25,
2. cos((x+y)/2) cos((x-y)/2) = √(3)/4.

Let's solve this system of equations.

Dividing the first equation by the second equation, we have:

(sin((x-y)/2) cos((x+y)/2)) / (cos((x+y)/2) cos((x-y)/2)) = (0.25) / (√(3)/4).

Simplifying the right side, we get:

(4 * sin((x-y)/2) cos((x+y)/2)) / (√(3) * cos((x+y)/2) cos((x-y)/2)) = 1 / √(3).

Canceling out the common terms, we have:

(4 * sin((x-y)/2)) / (√(3) * cos((x-y)/2)) = 1 / √(3).

Now, let's solve this equation for (x-y)/2:

4 * sin((x-y)/2) = (√(3) * cos((x-y)/2)).

Dividing both sides by cos((x-y)/2), we get:

4 * tan((x-y)/2) = √(3).

Taking the inverse tangent (arctan) of both sides, we have:

(x-y)/2 = arctan(√(3)/4).

Multiplying both sides by 2, we get:

x - y = 2 * arctan(√(3)/4).

Now, we have an equation relating x and y.

To find specific values for x and y, we would need additional information or constraints in the problem. Without any further information, we cannot determine unique values for x and y.