Question

Sin (4x-пі/3)=-1/2 розвязати рівняння​

Answer 1

$sin(4x-\frac{\pi}{3})=-\frac{1}{2}\\\\4x-\frac{\pi}{3}=(-1)^{n}\cdot (-\frac{\pi}{6})+\pi n=(-1)^{n+1}\cdot \frac{\pi}{6}+\pi n\; ,\; n\in Z\\\\4x=\frac{\pi}{3}+(-1)^{n+1}\cdot \frac{\pi}{6}+\pi n\; ,\; n\in Z\\\\x=\frac{\pi}{12}+(-1)^{n+1}\cdot \frac{\pi}{24}+\frac{\pi n}{4}\; ,\; n\in Z$

Answer 2

To solve the equation sin(4x - π/3) = -1/2, we need to find the values of x that satisfy this equation. To do that, we can follow these steps:

Step 1: Find the reference angle The reference angle is the positive acute angle between the terminal side of the angle (4x - π/3) and the x-axis. Since sin is negative in the third and fourth quadrants, we need to find the reference angle in either of these quadrants.

Reference angle = π/6 (30 degrees) [since sin(π/6) = 1/2]

Step 2: Set up the equation for the reference angle In the third quadrant, the angle (4x - π/3) has a reference angle of π/6, and the angle's terminal side is π - (π/6) = 5π/6.

So, we have: 4x - π/3 = 5π/6

Step 3: Solve for x Now, we'll solve for x:

4x = 5π/6 + π/3 4x = 5π/6 + 2π/6 4x = 7π/6

x = (7π/6) / 4 x = 7π/24

Step 4: Check other possible solutions Since sine is a periodic function, there can be other solutions for x within the given interval. The sine function repeats every 2π radians. So, we'll check for additional solutions by adding multiples of 2π:

x = 7π/24 + 2π(n) (where n is an integer)

These are the general solutions for the equation sin(4x - π/3) = -1/2.

If you need a specific numerical value, you can plug in values for n (e.g., n = 0, 1, -1, 2, -2, ...) to get different solutions for x. Remember that the solutions should fall within the desired interval.