Question

Sin (x+4пи/3) = 2 sin (4пи/3 - x) Sin x cos^2 x - 1/2sin^2 x - 1/4sin x + 3/8 =0

Answer 1

1) Вычислим  sin (4π/3)=sin (π+π/3)=sin π/3=√3/2 ,
                     сos (4π/3)=cos(π+π/3)=-сosπ/3=-1/2

по формулам приведения угол π+π/3 во второй четверти, синус во второй четверти имеет знак "+", косинус во второй четверти имеет знак "-"

Применим формулу синуса суммы к левой части уравнения и синуса разности к правой части уравнения:
sin x · cos (4π/3) + сos x · sin (4π/3) = 2 · sin (4π/3) · сos x -  2 · cos 4π/3 · sin x
Заменим sin (4π/3)=√3/2 , сos (4π/3)=-1/2, получим

-(sin x )/2 +(√3·cosx)/2=√3 сos x + sin x
или
√3 cos x +3 sinx =0
cos x и sin x одновременно равняться нулю не могут ( если один 0, то другой 1 или -1), поэтому делим уравнение на соsx≠0,
получаем 3 tg x=-√3    или    tg x= -√3/3,    х=- π/6 +π· k, k∈Z

2)  Заменим  sin²x=1-cos²x
sin x · cos² x - 1/2 (1-cos² x)-1/4 sin x +3/8=0,
Сгруппируем первое и третье, второе и четвертое:
sin x (cos²x - 1/4) + 1/2 ( cos²x- 1/4)=0
(cos² x - 1/4)(sin x +1/2)=0
или
(cosx-1/2) ( сos x+1/2)(sin x + 1/2)=0
Произведение трех множителей равно нулю, когда хотя бы один из них равен нулю:

$cosx= \frac{1}{2}\Rightarrow x=\pm \frac{ \pi }{3}+2 \pi k, k\in Z; \\ cosx=- \frac{1}{2} \Rightarrow x=\pm \frac{2 \pi }{3} +2 \pi n, n\in Z; \\ sinx=- \frac{1}{2}\Rightarrow x=(-1) ^{m+1}\cdot \frac{ \pi }{6} + \pi m, m\in Z.$

Ответ. $x=\pm \frac{ \pi }{3}+2 \pi k, k\in Z; \\ x=\pm \frac{2 \pi }{3} +2 \pi n, n\in Z;\\ x=(-1) ^{m+1}\cdot \frac{ \pi }{6} + \pi m, m\in Z.$
  

Answer 2

Let's analyze the given equations step by step.

1. Sin(x + 4π/3) = 2sin(4π/3 - x):

To simplify this equation, we can use the trigonometric identity: sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

Using this identity, we have: sin(x + 4π/3) = sin(x)cos(4π/3) + cos(x)sin(4π/3) sin(x + 4π/3) = sin(x)(-1/2) + cos(x)(√3/2)

Now, let's simplify the right side of the equation: 2sin(4π/3 - x) = 2sin(4π/3)cos(x) - 2cos(4π/3)sin(x) 2sin(4π/3 - x) = (√3/2)cos(x) - (1/2)sin(x)

Comparing the two sides of the equation, we have: sin(x)(-1/2) + cos(x)(√3/2) = (√3/2)cos(x) - (1/2)sin(x)

To simplify further, we can equate the coefficients of sin(x) and cos(x): -1/2 = -1/2 √3/2 = √3/2

Both sides of the equation are equal, so the equation is true for all values of x.

2. Sin(x)cos^2(x) - 1/2sin^2(x) - 1/4sin(x) + 3/8 = 0:

To solve this equation, we can factor it: sin(x)[cos^2(x) - 1/2sin(x) - 1/4] + 3/8 = 0

Now, let's focus on the expression in the square brackets: cos^2(x) - 1/2sin(x) - 1/4

We can rewrite sin(x) as 1 - cos^2(x) using the identity sin^2(x) + cos^2(x) = 1: cos^2(x) - 1/2(1 - cos^2(x)) - 1/4 cos^2(x) - 1/2 + 1/2cos^2(x) - 1/4 3/2cos^2(x) - 3/4 - 1/2 3/2cos^2(x) - 5/4

Now, let's substitute this expression back into the equation: sin(x)(3/2cos^2(x) - 5/4) + 3/8 = 0

This equation is not easily solvable using simple algebraic techniques. To find the solutions, you can use numerical methods or graphing tools to approximate the values of x that satisfy the equation.