The greatest integer less than or equal to `1/(cos290^0)+1/(sqrt(3)sin250^0)` is _____

Explanation

To solve the problem, we need to find the greatest integer less than or equal to the expression \( \frac{1}{\cos 290^\circ} + \frac{1}{\sqrt{3} \sin 250^\circ} \).

### Step-by-Step Solution:

1. **Convert the angles:**
- We know that \( \cos 290^\circ = \cos(270^\circ + 20^\circ) \) and \( \sin 250^\circ = \sin(270^\circ - 20^\circ) \).
- Using the angle subtraction and addition formulas:
\[ \cos 290^\circ = \sin 20^\circ \]
\[ \sin 250^\circ = -\cos 20^\circ \]

2. **Substitute the values into the expression:**
- Now substitute these values into the original expression:
\[ \frac{1}{\cos 290^\circ} + \frac{1}{\sqrt{3} \sin 250^\circ} = \frac{1}{\sin 20^\circ} + \frac{1}{\sqrt{3} (-\cos 20^\circ)} \]
- This simplifies to:
\[ \frac{1}{\sin 20^\circ} - \frac{1}{\sqrt{3} \cos 20^\circ} \]

3. **Finding a common denominator:**
- The common denominator for the two fractions is \( \sqrt{3} \sin 20^\circ \cos 20^\circ \):
\[ \frac{\sqrt{3} \cos 20^\circ - \sin 20^\circ}{\sqrt{3} \sin 20^\circ \cos 20^\circ} \]

4. **Using trigonometric identities:**
- We can rewrite the numerator:
\[ \sqrt{3} \cos 20^\circ - \sin 20^\circ = 2 \left( \frac{\sqrt{3}}{2} \cos 20^\circ - \frac{1}{2} \sin 20^\circ \right) \]
- Recognizing that \( \frac{\sqrt{3}}{2} = \sin 60^\circ \) and \( \frac{1}{2} = \cos 60^\circ \):
\[ = 2 \sin(60^\circ - 20^\circ) = 2 \sin 40^\circ \]

5. **Substituting back into the expression:**
- Now we have:
\[ \frac{2 \sin 40^\circ}{\sqrt{3} \sin 20^\circ \cos 20^\circ} \]
- Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \):
\[ \sin 40^\circ = 2 \sin 20^\circ \cos 20^\circ \]
- Thus, the expression simplifies to:
\[ \frac{2 \cdot 2 \sin 20^\circ \cos 20^\circ}{\sqrt{3} \sin 20^\circ \cos 20^\circ} = \frac{4}{\sqrt{3}} \]

6. **Finding the greatest integer less than or equal to the value:**
- Now we need to find the greatest integer less than or equal to \( \frac{4}{\sqrt{3}} \).
- Rationalizing gives:
\[ \frac{4 \sqrt{3}}{3} \approx 2.3094 \]
- The greatest integer less than or equal to \( 2.3094 \) is \( 2 \).

### Final Answer:
The greatest integer less than or equal to \( \frac{1}{\cos 290^\circ} + \frac{1}{\sqrt{3} \sin 250^\circ} \) is **2**.