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

We are given the expression \( \sqrt{3} \csc 20^\circ - \sec 20^\circ \), and we need to simplify it step by step. Step 1: Express the terms using basic trigonometric identities. We know the following basic trigonometric identities: \[ \csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}. \] Therefore, \[ \csc 20^\circ = \frac{1}{\sin 20^\circ}, \quad \sec 20^\circ = \frac{1}{\cos 20^\circ}. \] Substituting these into the original expression gives: \[ \sqrt{3} \csc 20^\circ - \sec 20^\circ = \sqrt{3} \times \frac{1}{\sin 20^\circ} - \frac{1}{\cos 20^\circ}. \] Step 2: Find an approximation for \( \sin 20^\circ \) and \( \cos 20^\circ \). Using known values or a calculator, we approximate: \[ \sin 20^\circ \approx 0.3420, \quad \cos 20^\circ \approx 0.9397. \] Substitute these values into the expression: \[ \sqrt{3} \times \frac{1}{0.3420} - \frac{1}{0.9397}. \] Step 3: Simplify the expression. Now, calculate the individual terms: \[ \sqrt{3} \approx 1.732, \] \[ \frac{1.732}{0.3420} \approx 5.06, \quad \frac{1}{0.9397} \approx 1.064. \] Thus, the expression becomes: \[ 5.06 - 1.064 = 4. \] Step 4: Conclusion. The value of \( \sqrt{3} \csc 20^\circ - \sec 20^\circ \) is approximately 4.