Question

Find all the solutions to

3+2sinx=4
.

Assume that n
is an integer.

π/6+2πn
π/6+2πn and 5π/6+2πn
There are no solutions to the equation.
π/3+2πn and 5π/3+2πn

Answer 1

To solve the equation \( 3 + 2\sin x = 4 \), we can start by isolating the sine function:

\[
2\sin x = 4 - 3
\]
\[
2\sin x = 1
\]
\[
\sin x = \frac{1}{2}
\]

Next, we need to find all values of \( x \) for which \( \sin x = \frac{1}{2} \). The sine function equals \( \frac{1}{2} \) at the angles:

\[
x = \frac{\pi}{6} + 2\pi n \quad \text{and} \quad x = \frac{5\pi}{6} + 2\pi n
\]

for any integer \( n \).

Thus, the complete set of solutions is:

\[
x = \frac{\pi}{6} + 2\pi n \quad \text{and} \quad x = \frac{5\pi}{6} + 2\pi n
\]

where \( n \) is any integer.

Therefore, the correct answer is:

**\(\frac{\pi}{6} + 2\pi n\) and \(\frac{5\pi}{6} + 2\pi n\)**.