Question

Evaluate: \[ \frac{1}{\cos 290^\circ} + \frac{1}{\sqrt{3} \sin 250^\circ} \]

• A. \( \frac{\sqrt{3}}{4} \)
• B. \( \frac{4}{\sqrt{3}} \)
• C. \( \frac{2}{\sqrt{3}} \)
• D. \( \frac{\sqrt{3}}{2} \)

Hint:

When simplifying trigonometric expressions, use standard angle values or identities to relate terms to simpler trigonometric functions.

Answer 1

The Correct Option is B

We are given: \[ \frac{1}{\cos 290^\circ} + \frac{1}{\sqrt{3} \sin 250^\circ} \] Step 1: Calculate \( \cos 290^\circ \) and \( \sin 250^\circ \). - \( \cos 290^\circ = \cos (360^\circ - 70^\circ) = \cos 70^\circ \), and from standard trigonometric values, \( \cos 70^\circ = \sin 20^\circ \). - \( \sin 250^\circ = \sin (270^\circ - 20^\circ) = -\cos 20^\circ \). Step 2: Substituting these values back into the expression: \[ \frac{1}{\cos 290^\circ} + \frac{1}{\sqrt{3} \sin 250^\circ} = \frac{1}{\sin 20^\circ} + \frac{1}{-\sqrt{3} \cos 20^\circ} \] Step 3: Simplify the expression. Since \( \sin 20^\circ \approx 0.342 \) and \( \cos 20^\circ \approx 0.94 \), we can further simplify: \[ \frac{1}{\cos 290^\circ} + \frac{1}{\sqrt{3} \sin 250^\circ} \approx \frac{4}{\sqrt{3}} \] % Final Answer \[ \boxed{\frac{4}{\sqrt{3}}} \]