Question

If \(2\tan A = 3\), then value of \(\sec A\) equals

• A. \(\frac{\sqrt{13}}{2}\)
• B. \(\frac{\sqrt{13}}{4}\)
• C. \(\frac{2}{\sqrt{13}}\)
• D. \(\frac{\sqrt{13}}{2}\)

Hint:

Alternatively, use a right triangle. If \(\tan A = 3/2\), the opposite side is 3 and adjacent side is 2. The hypotenuse is \(\sqrt{3^2 + 2^2} = \sqrt{13}\). Thus, \(\sec A = \text{hypotenuse} / \text{adjacent} = \sqrt{13}/2\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Tangent and Secant are related by the fundamental identity \(\sec^2 A = 1 + \tan^2 A\).
Step 2: Key Formula or Approach:
From \(2\tan A = 3\), we get \(\tan A = \frac{3}{2}\).
Step 3: Detailed Explanation:
Using the identity:
\[ \sec^2 A = 1 + \tan^2 A \]
\[ \sec^2 A = 1 + \left( \frac{3}{2} \right)^2 \]
\[ \sec^2 A = 1 + \frac{9}{4} \]
\[ \sec^2 A = \frac{4 + 9}{4} = \frac{13}{4} \]
Taking the square root:
\[ \sec A = \frac{\sqrt{13}}{\sqrt{4}} = \frac{\sqrt{13}}{2} \]
Step 4: Final Answer:
The value of \(\sec A\) is \(\frac{\sqrt{13}}{2}\).