Question

Cos(2x-2п/3)+cos(x-п/3)=0

Answer 1

Ответ:

$\frac{ - \cos(2x) + \sqrt{3} \times \sin(2x) }{2} + \cos( \frac{x - \pi}{3} )$

Объяснение:

1) Разложить выражение:

$\cos(2x) \cos( \frac{2\pi}{3} ) + \sin(2x) \sin( \frac{2\pi}{3} ) + \cos( \frac{x\pi}{3} )$

2) Вычислить, используя таблицу значений тригонометрических функций:

$\cos(2x) \times ( - \frac{1}{2} ) + \sin(2x) \times \frac{ \sqrt{3} }{2} + \cos( \frac{x - \pi}{3} )$

3) Вычислить:

$- \frac{ \cos(2x) }{2} + \frac{ \sqrt{3} \times \sin(2x) }{2} + \cos( \frac{x - \pi}{3} )$

4) Записать все числители над общим знаменателем:

$\frac{ - \cos(2x) + \sqrt{3} \times \sin(2x) }{2} + \cos( \frac{x - \pi}{3} )$

Answer 2

To solve the trigonometric equation cos(2x - 2π/3) + cos(x - π/3) = 0, we can use trigonometric identities to simplify the equation. The sum-to-product identity is helpful in this case:

cos(A) + cos(B) = 2 * cos((A + B) / 2) * cos((A - B) / 2)

Let's apply this identity to our equation:

cos(2x - 2π/3) + cos(x - π/3) = 0

2 * cos((2x - 2π/3 + x - π/3) / 2) * cos((2x - 2π/3 - x + π/3) / 2) = 0

2 * cos((3x - π) / 2) * cos((x - π) / 2) = 0

Now, we have two separate cases to consider when the product is equal to zero:

1. cos((3x - π) / 2) = 0
2. cos((x - π) / 2) = 0

Case 1: cos((3x - π) / 2) = 0

To find the solutions for this case, we solve for (3x - π) / 2:

(3x - π) / 2 = π/2 + kπ where k is an integer

3x - π = π + 2kπ

3x = 2π + 2kπ

x = (2π + 2kπ) / 3

x = (2 + 2k)π / 3

Case 2: cos((x - π) / 2) = 0

To find the solutions for this case, we solve for (x - π) / 2:

(x - π) / 2 = π/2 + kπ where k is an integer

x - π = π + 2kπ

x = 2π + 2kπ

x = (2 + 2k)π

So, the general solutions for the equation are:

1. x = (2 + 2k)π / 3
2. x = (2 + 2k)π

where k is an integer. These solutions cover all possible values of x that satisfy the given equation.