Question

( x + y = π\3, sin x + sin y = 1. пожалуйста помогите решить

Answer 1

Выразим у из первого уравнения у=π/3 - х 
и подставим во второе 
 sin x + sin (π/3 - х )= 1. 
Применяем формулу
sinα+sinβ=2·sin((α+β)/2)·cos((α-β)/2)

2·sin ((x+π/3 - х )/2)  ·   cos( (x - π/3 + х )/2)=1

2·sin (π/6)  ·  cos (x-   (π/6))=1

2·(1/2) ·  cos (x-   (π/6))=1
 
 cos (x-   (π/6))=1
 x-   (π/6)= 2πn, n∈ Z

x=(π/6) + 2πn,  n∈ Z

у=(π/3) -  х=  (π/3)   -   (π/6) -   2πn,    n∈Z

Ответ. ((π/6) + 2πn;   (π/6) -   2πn,    n∈Z)

Answer 2

Solving the Equations

To solve the equations x + y = π/3 and sin(x) + sin(y) = 1, we can use various mathematical techniques such as substitution, elimination, or graphing. Let's start by rearranging the first equation to express y in terms of x and then substitute it into the second equation to solve for x.

Using Substitution

We can express y in terms of x from the first equation: y = π/3 - x. Then, substitute this expression for y into the second equation: sin(x) + sin(π/3 - x) = 1.

Solving for x

We can then solve for x using trigonometric identities and algebraic manipulation. Once we find the value of x, we can substitute it back into the equation x + y = π/3 to find the value of y.

Let's proceed with these steps to find the values of x and y.