Question

Sin(6x-п/3)=sin(2x+п/4)

Answer 1

To solve the trigonometric equation sin(6x - π/3) = sin(2x + π/4), we can use the trigonometric identity for sine:

sin(A) = sin(B) if and only if (A = B + 2πn) or (A = π - B + 2πn)

where n is an integer.

First, let's equate the arguments of sine in the given equation:

6x - π/3 = 2x + π/4

Now, we'll solve for x:

4x = π/4 + π/3 4x = (3π + 4π) / 12 4x = 7π / 12 x = (7π / 12) / 4 x = 7π / (4 * 12) x = 7π / 48

Now, we can find additional solutions using the trigonometric identity mentioned earlier. For any integer n, we have:

x = (7π / 48) + 2π * n x = (7π + 96π * n) / 48 x = (7π + 96π * n) / 48 x = (7 + 96 * n)π / 48

So, the general solution to the equation is:

x = (7 + 96 * n)π / 48, where n is an integer.