Solve the education : sin(3x+30) =√3/2

To solve the equation ( \sin(3x + 30^\circ) = \frac{\sqrt{3}}{2} ), we need to find the angles where the sine value is equal to ( \frac{\sqrt{3}}{2} ).

The sine function equals ( \frac{\sqrt{3}}{2} ) at:

[ 3x + 30^\circ = 60^\circ + k \cdot 360^\circ \quad \text{and} \quad 3x + 30^\circ = 120^\circ + k \cdot 360^\circ ]

where ( k ) is any integer.

Solving the first equation:

1. Equation: [ 3x + 30^\circ = 60^\circ + k \cdot 360^\circ ] 2. Rearranging: [ 3x = 30^\circ + k \cdot 360^\circ ] 3. Dividing by 3: [ x = 10^\circ + k \cdot 120^\circ ]

Solving the second equation:

1. Equation: [ 3x + 30^\circ = 120^\circ + k \cdot 360^\circ ] 2. Rearranging: [ 3x = 90^\circ + k \cdot 360^\circ ] 3. Dividing by 3: [ x = 30^\circ + k \cdot 120^\circ ]

Final Solutions:

Putting it together, we have two families of solutions:

1. From the first set: [ x = 10^\circ + k \cdot 120^\circ, \quad k \in \mathbb{Z} ]

2. From the second set: [ x = 30^\circ + k \cdot 120^\circ, \quad k \in \mathbb{Z} ]

These equations represent the general solutions for ( x ). Depending on the specific interval you are interested in, you can choose integer values for ( k ) to find particular solutions.