Question

Find all solutions of cos(3θ/2)=1/2
on [0,2π).



θ=2π/9,2π/3,10π/9,14π/9
θ=2π/9,10π/9,14π/9
θ=4π/9,8π/9,4π/3,16π/9
θ=2π/9,2π/3,10π/9
There are no solutions to this equation.

Answer 1

To solve the equation \(\cos\left(\frac{3\theta}{2}\right) = \frac{1}{2}\), we need to find the angles whose cosine equals \(\frac{1}{2}\).

The cosine function equals \(\frac{1}{2}\) at the following angles in the interval \([0, 2\pi)\):
- \(\frac{\pi}{3}\)
- \(\frac{5\pi}{3}\)

Thus, we have:

\[
\frac{3\theta}{2} = \frac{\pi}{3} + 2k\pi \quad \text{for integer } k
\]
\[
\frac{3\theta}{2} = \frac{5\pi}{3} + 2k\pi \quad \text{for integer } k
\]

Now, we will solve for \(\theta\) in each case.

**Case 1:**

Starting with the first equation:

\[
\frac{3\theta}{2} = \frac{\pi}{3}
\]

Multiply both sides by \(\frac{2}{3}\):

\[
\theta = \frac{2}{3} \cdot \frac{\pi}{3} = \frac{2\pi}{9}
\]

Adding \(2k\pi\) gives:

\[
\theta = \frac{2\pi}{9} + \frac{4k\pi}{3}
\]

To find values in \([0, 2\pi)\):

- For \(k = 0\): \(\theta = \frac{2\pi}{9}\)
- For \(k = 1\): \(\theta = \frac{2\pi}{9} + \frac{4\pi}{3} = \frac{2\pi}{9} + \frac{12\pi}{9} = \frac{14\pi}{9}\)

- For \(k = -1\): \(\theta = \frac{2\pi}{9} - \frac{4\pi}{3} = \frac{2\pi}{9} - \frac{12\pi}{9} = \frac{-10\pi}{9}\) (not in range)

Valid solutions from this case: \(\frac{2\pi}{9}, \frac{14\pi}{9}\)

**Case 2:**

Now for the second equation:

\[
\frac{3\theta}{2} = \frac{5\pi}{3}
\]

Multiply both sides by \(\frac{2}{3}\):

\[
\theta = \frac{2}{3} \cdot \frac{5\pi}{3} = \frac{10\pi}{9}
\]

Adding \(2k\pi\) gives:

\[
\theta = \frac{10\pi}{9} + \frac{4k\pi}{3}
\]

To find values in \([0, 2\pi)\):

- For \(k = 0\): \(\theta = \frac{10\pi}{9}\)
- For \(k = 1\): \(\theta = \frac{10\pi}{9} + \frac{4\pi}{3} = \frac{10\pi}{9} + \frac{12\pi}{9} = \frac{22\pi}{9}\) (not in range)

- For \(k = -1\): \(\theta = \frac{10\pi}{9} - \frac{4\pi}{3} = \frac{10\pi}{9} - \frac{12\pi}{9} = \frac{-2\pi}{9}\) (not in range)

Valid solutions from this case: \(\frac{10\pi}{9}\)

### Final Solutions

Combining all valid \(\theta\) values, we get:

\[
\theta = \frac{2\pi}{9}, \frac{10\pi}{9}, \frac{14\pi}{9}
\]

Thus, the complete set of solutions is:

\[
\boxed{\frac{2\pi}{9}, \frac{10\pi}{9}, \frac{14\pi}{9}}
\]