Question

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

Answer 1

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.