Тригонометрия (системы). Решите пожалуйста 647, по возможности 648. Спасибо!. Created by kartew. algebra-ru

Тригонометрия (системы). Решите пожалуйста 647, по возможности 648. Спасибо!...

kartew @kartew

August 2021 1 56 Report

Тригонометрия (системы). Решите пожалуйста 647, по возможности 648. Спасибо!

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oganesbagoyan

Verified answer

Task/28423814
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647. а)
{ cosxcosy =1/4;    { cosxcosy+sinxsiny =1/4+1/4 ;    { cos(x-y) =1/2  ;
< ⇔ <    ⇔  <
{ sinxsiny = 1/4. { cosxcosy-sinxsiny =1/4-1/4. { cos(x+y) = 0.
---
{ x - y = ± π/3 +2πm , m ∈ ℤ
<
{ x + y = π/2 +πl , l  ∈ ℤ .
-----------
a)
{ x - y = π/3 +2πm , m ∈ ℤ     { 2x = (π/3+π/2) + πl+2πm ,   m,l ∈ ℤ
< ⇔ <
{ x+y = π/2 +πl , l ∈ ℤ .    { 2y = (π/2-π/3)+ πl -2πm,   m,l ∈ ℤ .

{ x = 5π/12+ (l+2m )π/2 ,   m,l ∈ ℤ { x = 5π/12+(π/2)*k ,   k ∈ ℤ
< ⇔   <
{ y = π/12 + (l - 2m)π/2, m,l ∈ ℤ .    { y = π/12 +(π/2)*n,   n ∈ ℤ .
l+2m=k ,   l - 2m =n
-----------------
б)
{ x - y = - π/3 +2πm , m ∈ ℤ     { 2x = (π/2 - π/3)+ πl+2πm ,   m,l ∈ ℤ
< ⇔ <
{ x+ y = π/2 +πl , m ∈ ℤ .     { 2y = (π/2  +π/3)+ πl - 2πm, m,l ∈ ℤ .

  { x = π/12+  (π/2)*(l+2m ) ,   m,l ∈ ℤ { x = π/12+ (π/2)*k ,   k ∈ ℤ
< ⇔  <
  { y = 5π/12 + (π/2)*(l -2m), m,l ∈ ℤ .   { y = 5π/12 +(π/2)*n , n ∈ ℤ .
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
647. б)
{tgx*tgy =3 , { sinxsiny / cosxcosy = 3 , {cosxcosy =1/4 ,
< ⇔ < ⇔  < || ± || ⇔
{ sinxsiny =3/4. { sinxsiny =3/4.   {sinxsiny = 3/4.
---
{cosxcosy -sinxsiny =1/4-3/4, {cos(x+y) =-1/2, { x+y =±2π/3+2πm , m∈ ℤ
<   ⇔ < ⇔ <
{cosxcosy +sinxsiny =1/4 + 3/4. {cos(x-y) =1.   { x-y =2πl ,l ∈ ℤ
---
{2x  = ±2π/3 +2(m +l) , m,l ∈ ℤ    { x = ±π/3+π(m+l) ,   m,l ∈ ℤ
< ⇔   <
{ 2y =  ±2π/3+2(m-l)π   ,m,l ∈ ℤ . { y = ±π/3+π(m-l) ,   m,l ∈ ℤ .
* * * m+l= k ∈ ℤ ,   m - l =n∈ ℤ . * * *
{ x = ±π/3+πk  ,   k ∈ ℤ
<
{ y =± π/3 + πn,   n ∈ ℤ .

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kartew спасибо) Вы меня не впервые выручаете))

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