Question

The value of \( \sqrt{3} \csc 20^\circ - \sec 20^\circ \) is:

• A. 4
• B. 2
• C. 3
• D. 1

Hint:

When solving trigonometric expressions, it's often useful to approximate the values of sine and cosine, then simplify the expression step by step.

Answer 1

The Correct Option is D

The expression \( \sqrt{3} \csc 20^\circ - \sec 20^\circ \) is to be simplified. Step 1: Rewrite using reciprocal identities. The given expression is \( \sqrt{3} \csc 20^\circ - \sec 20^\circ \). Using the identities \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \), we get: \( \sqrt{3} \times \frac{1}{\sin 20^\circ} - \frac{1}{\cos 20^\circ} \). Step 2: Substitute approximate values. Approximate values for \( \sin 20^\circ \) and \( \cos 20^\circ \) are \( 0.3420 \) and \( 0.9397 \), respectively. The expression becomes: \( \sqrt{3} \times \frac{1}{0.3420} - \frac{1}{0.9397} \). Step 3: Perform calculations. Approximate \( \sqrt{3} \approx 1.732 \). Calculate the terms: \( \frac{1.732}{0.3420} \approx 5.06 \) \( \frac{1}{0.9397} \approx 1.064 \) Subtract the terms: \( 5.06 - 1.064 = 4 \). Step 4: Final Result. The simplified value of \( \sqrt{3} \csc 20^\circ - \sec 20^\circ \) is approximately 4.