Question

When \(\sin A = \frac{1}{3}\), the value of \(\cot A\) is

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

Hint:

You can also solve this problem quickly using trigonometric identities:
\[ \csc A = \frac{1}{\sin A} = 3 \] Using the identity \(\cot^2 A = \csc^2 A - 1\):
\[ \cot^2 A = 3^2 - 1 = 9 - 1 = 8 \] \[ \cot A = \sqrt{8} = 2\sqrt{2} \] Using algebraic identities avoids drawing triangles and is much faster!

Answer 1

The Correct Option is B

Step 1: Find cos A using the Pythagorean identity.
Given \(\sin A = \frac{1}{3}\), use \(\cos^2 A = 1 - \sin^2 A\).
Step 2: Substitute and simplify.
\[ \cos^2 A = 1 - \frac{1}{9} = \frac{8}{9} \implies \cos A = \frac{2\sqrt{2}}{3} \] (A is acute here, so cos A is taken positive.)
Step 3: Form cot A as the ratio of cos A to sin A.
\[ \cot A = \frac{\cos A}{\sin A} = \frac{\frac{2\sqrt{2}}{3}}{\frac{1}{3}} \]
Step 4: Simplify to get the final value.
\[ \cot A = 2\sqrt{2} \]
This matches option (B).
\[ \boxed{\cot A = 2\sqrt{2}} \]