Question

If \( 2 \sin A = 1 \), then the value of \( \tan A + \cot A \) is :

• A. \( \sqrt{3} \)
• B. \( \frac{4}{\sqrt{3}} \)
• C. \( \frac{\sqrt{3}}{2} \)
• D. \( 1 \)

Hint:

\( \tan \theta + \cot \theta = \frac{1}{\sin \theta \cos \theta} \). This identity is also useful for faster calculations.

Answer 1

The Correct Option is B

To solve this problem, we start with the given equation:

\(2 \sin A = 1\)

From the equation above, we can find the value of \(\sin A\):

\(\sin A = \frac{1}{2}\)

We know that \(\sin A = \frac{1}{2}\) when \(A = 30^\circ\) (or \(\frac{\pi}{6}\) radians) because \(\sin 30^\circ = \frac{1}{2}\).

Next, we calculate \(\tan A\) and \(\cot A\):

The formula for tangent is \(\tan A = \frac{\sin A}{\cos A}\). From trigonometric identities, \(\cos 30^\circ = \frac{\sqrt{3}}{2}\).

• \(\tan 30^\circ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}\)

Similarly, the cotangent is the reciprocal of tangent, i.e., \(\cot A = \frac{1}{\tan A}\).

• \(\cot 30^\circ = \sqrt{3}\)

Now, we find the sum \(\tan A + \cot A\):

\(\tan 30^\circ + \cot 30^\circ = \frac{1}{\sqrt{3}} + \sqrt{3}\)

To simplify, get a common denominator:

\(\frac{1}{\sqrt{3}} + \frac{3}{\sqrt{3}} = \frac{4}{\sqrt{3}}\)

Thus, the value of \(\tan A + \cot A\) is \(\frac{4}{\sqrt{3}}\).

Therefore, the correct answer is:

\(\frac{4}{\sqrt{3}}\)