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