Find the values of `theta` between `0^(@)` and `360^(@)` which satisfy the equations
`cos2theta=(1)/(2)`.

Answer

To solve the equation \( \cos 2\theta = \frac{1}{2} \) for values of \( \theta \) between \( 0^\circ \) and \( 360^\circ \), we will follow these steps:

### Step 1: Identify the general solution for \( \cos x = \frac{1}{2} \)

The cosine function equals \( \frac{1}{2} \) at specific angles. We know that:
\[ \cos x = \frac{1}{2} \implies x = 60^\circ + 360^\circ n \quad \text{or} \quad x = 300^\circ + 360^\circ n \quad (n \in \mathbb{Z}) \]

### Step 2: Substitute \( 2\theta \) for \( x \)

Since we have \( \cos 2\theta = \frac{1}{2} \), we can set:
\[ 2\theta = 60^\circ + 360^\circ n \quad \text{or} \quad 2\theta = 300^\circ + 360^\circ n \]

### Step 3: Solve for \( \theta \)

Now, we will solve for \( \theta \) in both cases.

**Case 1:**
\[ 2\theta = 60^\circ \implies \theta = \frac{60^\circ}{2} = 30^\circ \]

**Case 2:**
\[ 2\theta = 300^\circ \implies \theta = \frac{300^\circ}{2} = 150^\circ \]

### Step 4: Consider additional solutions

Since \( n \) can be any integer, we also consider \( n = 1 \) for both cases:

**For \( n = 1 \):**
- From \( 2\theta = 60^\circ + 360^\circ \cdot 1 \):
\[ 2\theta = 420^\circ \implies \theta = \frac{420^\circ}{2} = 210^\circ \]

- From \( 2\theta = 300^\circ + 360^\circ \cdot 1 \):
\[ 2\theta = 660^\circ \implies \theta = \frac{660^\circ}{2} = 330^\circ \]

### Step 5: Collect all solutions

The values of \( \theta \) that satisfy \( \cos 2\theta = \frac{1}{2} \) within the range \( 0^\circ \) to \( 360^\circ \) are:
\[ \theta = 30^\circ, 150^\circ, 210^\circ, 330^\circ \]

### Final Answer:
The values of \( \theta \) are \( 30^\circ, 150^\circ, 210^\circ, \) and \( 330^\circ \).

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