Question

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

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

Hint:

Using the identity \(\sec A = \sqrt{1 + \tan^2 A}\) is much faster than drawing a right-angled triangle and avoids potential arithmetic errors.
Always keep basic Pythagorean identities memorized for speed!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The topic of this question is Trigonometric Identities and Ratios.
We are given a trigonometric relation \(2 \tan A = 3\), which simplifies to \(\tan A = \frac{3}{2}\).
We need to determine the value of \(\sec A\) using trigonometric relationships.

Step 2: Key Formula or Approach:
We can solve this using either of two methods:
1. Apply the fundamental Pythagorean identity relating secant and tangent:
\[ \sec^2 A = 1 + \tan^2 A \implies \sec A = \sqrt{1 + \tan^2 A} \] 2. Draw a right-angled triangle where \(\tan A = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{3}{2}\), use the Pythagoras theorem to find the hypotenuse, and then find \(\sec A = \frac{\text{Hypotenuse}}{\text{Adjacent}}\).

Step 3: Detailed Explanation:

• Simplify the given equation:
\[ 2 \tan A = 3 \implies \tan A = \frac{3}{2} \]

• Apply the Pythagorean identity:
\[ \sec^2 A = 1 + \tan^2 A \]

• Substitute the value of \(\tan A\):
\[ \sec^2 A = 1 + \left(\frac{3}{2}\right)^2 \] \[ \sec^2 A = 1 + \frac{9}{4} \]

• Find a common denominator to add the terms:
\[ \sec^2 A = \frac{4 + 9}{4} = \frac{13}{4} \]

• Take the square root on both sides to find \(\sec A\):
\[ \sec A = \sqrt{\frac{13}{4}} = \frac{\sqrt{13}}{2} \]

Step 4: Final Answer:
The value of \(\sec A\) is \(\frac{\sqrt{13}}{2}\).
Therefore, the correct option is (D).