Question

sin(2x+pi/3)cos(2x+pi/3)=0решите пожалуйста

Answer 1

Произведение равно нулю, если хотябы один из множителей равен нулю.

Тоесть sin(2x+pi/3)=0 и cos(2x+pi/3)=0. Решу каждое по отдельности.

$sin(2x+\frac{\pi}{3})=0\\2x+\frac{\pi}{3}=\pi*n\\2x=\pi*n-\frac{\pi}{3}\\x=\frac{\pi*n}{2}-\frac{\pi}{6}$

n принадлежит Z.

 

 $cos(2x+\frac{\pi}{3})=0\\2x+\frac{\pi}{3}=\frac{\pi}{2}+\pi*k\\2x=\frac{3\pi}{6}-\frac{2\pi}{6}+\pi*k\\2x=\frac{\pi}{6}+\pi*k\\x=\frac{\pi}{12}+\frac{\pi*k}{2}$

k принадлежит Z. 

 

Answer 2

Solution for sin(2x+pi/3)cos(2x+pi/3)=0

To solve the equation sin(2x+π/3)cos(2x+π/3) = 0, we can use the trigonometric identities for sine and cosine to find the values of x that satisfy the equation.

1. Using Trigonometric Identities: - We can use the trigonometric identity sin(2θ) = 2sin(θ)cos(θ) to simplify the equation. - Applying this identity to sin(2x+π/3), we get 2sin(x+π/6)cos(x+π/6)cos(2x+π/3) = 0.

[[9 #1]]

2. Finding Solutions: - Now, we can solve for x using the fact that the product of two numbers is zero if and only if at least one of the numbers is zero. - This gives us two cases to consider: - Case 1: 2sin(x+π/6)cos(x+π/6) = 0 - Case 2: cos(2x+π/3) = 0

3. Solving Case 1: - For 2sin(x+π/6)cos(x+π/6) = 0, we can use the identity 2sin(θ)cos(θ) = sin(2θ) to simplify further. - This gives us sin(2x+π/3) = 0.

- Solving sin(2x+π/3) = 0 gives us the solutions: - x = nπ, where n is an integer. - x = ±π/3 + 2nπ, where n is an integer.

[[9 #2]]

4. Solving Case 2: - For cos(2x+π/3) = 0, we can solve for x by finding the values that make cos(2x+π/3) equal to zero.

- Solving cos(2x+π/3) = 0 gives us the solutions: - 2x+π/3 = ±π/2 + 2nπ, where n is an integer. - 2x+π/3 = ±3π/2 + 2nπ, where n is an integer.

- Simplifying these equations gives us: - x = -π/6 + πn, where n is an integer. - x = -5π/6 + πn, where n is an integer.

[[9 #2]]

Conclusion:

The solutions to the equation sin(2x+π/3)cos(2x+π/3) = 0 are: - x = nπ, where n is an integer. - x = ±π/3 + 2nπ, where n is an integer. - x = -π/6 + πn, where n is an integer. - x = -5π/6 + πn, where n is an integer.