Question

Sin(x+п/6)+sin(x-п/6)=1/2

Answer 1

Sin(x+π/6)+sin(x-π/6)=0

2*sin(x+π/6+x-π/6)/2)*cos(x+π/6-x+π/6)/2

=1/2

2*sinx *cosπ/6=1/2

2*sinx*√3/2=1/2

sinx=1/(2√3)

sinx=√3/6

x=(-1)ⁿarcsin(√3/6)+πn;n€Z

Answer 2

To solve the equation sin(x + π/6) + sin(x - π/6) = 1/2, we can use the trigonometric identity for the sum of two sines:

sin(A + B) + sin(A - B) = 2 * sin(A) * cos(B).

Let A = x and B = π/6. Now, the equation becomes:

2 * sin(x) * cos(π/6) = 1/2.

The value of cos(π/6) is √3/2. Substituting this in:

2 * sin(x) * (√3/2) = 1/2.

Now, isolate sin(x):

sin(x) = (1/2) / (2 * √3/2) = (1/2) / √3 = 1 / (2√3).

To solve for x, take the arcsin (inverse sine) of both sides:

x = arcsin(1 / (2√3)).

Using a calculator, we can find the approximate value of x:

x ≈ 0.268 radians.

Keep in mind that sine has multiple solutions, as it is a periodic function. The general solution for x can be expressed as follows:

x = πn + (-1)^n * arcsin(1 / (2√3)),

where n is an integer (..., -2, -1, 0, 1, 2, ...). This form represents all possible solutions for x, considering the periodicity of the sine function.