$$ \cos x ^ { \circ } = \frac { 29 } { 30 } $$

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To solve the equation:

$$
\cos x^{\circ} = \frac{29}{30}
$$

we aim to find all real numbers $$ x $$ (in degrees) that satisfy this equation. Here's a step-by-step solution:

### 1. **Understanding the Equation**

The equation states that the cosine of angle $$ x $$ (measured in degrees) is equal to $$ \frac{29}{30} $$, which is approximately 0.9667.

### 2. **Finding the Principal Solution**

To find the principal value of $$ x $$, we'll use the inverse cosine function:

$$
x = \cos^{-1}\left(\frac{29}{30}\right)
$$

Using a calculator:

$$
x \approx \cos^{-1}(0.9667) \approx 14.997^\circ
$$

So, the principal solution is approximately:

$$
x \approx 15.0^\circ
$$

### 3. **Finding All Solutions Within One Full Rotation**

The cosine function is positive in both the **first** and **fourth** quadrants. Therefore, within the range $$ 0^\circ $$ to $$ 360^\circ $$, there are two solutions:

1. **First Quadrant:**
   $$
   x_1 \approx 15.0^\circ
   $$

2. **Fourth Quadrant:**
   $$
   x_2 = 360^\circ - x_1 \approx 360^\circ - 15.0^\circ = 345.0^\circ
   $$

### 4. **General Solution**

Since the cosine function is periodic with a period of $$ 360^\circ $$, the general solutions can be expressed as:

$$
x = \pm 15.0^\circ + 360^\circ \times k
$$

where $$ k $$ is any integer ($$ k \in \mathbb{Z} $$).

### 5. **Summary of Solutions**

- **Principal Solutions:**
  $$
  x \approx 15.0^\circ \quad \text{and} \quad x \approx 345.0^\circ
  $$

- **General Solutions:**
  $$
  x = 15.0^\circ + 360^\circ \times k \quad \text{or} \quad x = 345.0^\circ + 360^\circ \times k \quad \text{for any integer } k
  $$

### 6. **Verification**

To ensure the solutions are correct, you can plug them back into the original equation:

$$
\cos(15.0^\circ) \approx 0.9659 \approx \frac{29}{30} \quad \text{(since } \frac{29}{30} \approx 0.9667\text{)}
$$

$$
\cos(345.0^\circ) \approx \cos(-15.0^\circ) \approx 0.9659 \approx \frac{29}{30}
$$

Both values are approximately equal, validating our solutions.

---

**Final Answer:**

All real numbers congruent to approximately 15° or 345° modulo 360°. In other words, the solutions to cos x° = 29⁄30 are

x ≈ 15.0° + 360°·k

and

x ≈ 345.0° + 360°·k

for any integer k.
The solutions are approximately $$ x \approx 15^\circ $$ and $$ x \approx 345^\circ $$.

\( \cos x ^ { \circ } = \frac { 29 } { 30 } \)

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