Expression `(1)/(cos 290^(@)) + (1)/(sqrt(3) sin 250^(@))` equals

Options

A. `(sqrt(3))/(4)`
B. `(4)/(sqrt(3))`
C. `(2)/(sqrt(3))`
D. `(sqrt(3))/(2)`

Answer

To solve the expression \(\frac{1}{\cos 290^\circ} + \frac{1}{\sqrt{3} \sin 250^\circ}\), we will follow these steps:

### Step 1: Simplify the Trigonometric Functions
First, we simplify \(\cos 290^\circ\) and \(\sin 250^\circ\).

\[ \cos 290^\circ = \cos(270^\circ + 20^\circ) = -\sin 20^\circ \]
\[ \sin 250^\circ = \sin(270^\circ - 20^\circ) = -\cos 20^\circ \]

### Step 2: Substitute the Values
Now substitute these values back into the 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:

\[ -\frac{1}{\sin 20^\circ} - \frac{1}{\sqrt{3} \cos 20^\circ} \]

### Step 3: Find a Common Denominator
The common denominator for the two fractions is \(\sqrt{3} \sin 20^\circ \cos 20^\circ\). Thus, we rewrite the expression:

\[ -\frac{\sqrt{3} \cos 20^\circ + \sin 20^\circ}{\sqrt{3} \sin 20^\circ \cos 20^\circ} \]

### Step 4: Factor the Numerator
Now, we can factor the numerator:

\[ -\left(\sqrt{3} \cos 20^\circ + \sin 20^\circ\right) \]

### Step 5: Use the Sine Difference Identity
We can recognize that \(\sqrt{3} \cos 20^\circ + \sin 20^\circ\) can be expressed using the sine of a difference:

\[ \sqrt{3} \cos 20^\circ + \sin 20^\circ = 2 \left(\frac{\sqrt{3}}{2} \cos 20^\circ + \frac{1}{2} \sin 20^\circ\right) = 2 \sin(60^\circ + 20^\circ) = 2 \sin 80^\circ \]

### Step 6: Substitute Back
Now substituting this back into our expression gives:

\[ -\frac{2 \sin 80^\circ}{\sqrt{3} \sin 20^\circ \cos 20^\circ} \]

### Step 7: Use the Double Angle Formula
Using the double angle formula for sine, \(\sin 2a = 2 \sin a \cos a\):

\[ \sin 20^\circ \cos 20^\circ = \frac{1}{2} \sin 40^\circ \]

Thus, we have:

\[ -\frac{2 \sin 80^\circ}{\sqrt{3} \cdot \frac{1}{2} \sin 40^\circ} = -\frac{4 \sin 80^\circ}{\sqrt{3} \sin 40^\circ} \]

### Step 8: Final Simplification
Since \(\sin 80^\circ = \cos 10^\circ\) and \(\sin 40^\circ = \cos 50^\circ\):

\[ -\frac{4 \cos 10^\circ}{\sqrt{3} \cos 50^\circ} \]

### Conclusion
Thus, the final expression simplifies to:

\[ -\frac{4 \cos 10^\circ}{\sqrt{3} \cos 50^\circ} \]