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: Understanding the Question:
The topic of this question is Introduction to Trigonometry.
Trigonometric ratios represent the relationships between the acute angles of a right-angled triangle and the lengths of its sides.
We are given the sine of an angle \(A\) as \(\sin A = \frac{1}{3}\).
Our objective is to find the value of \(\cot A\) using either a right-angled triangle or fundamental trigonometric identities.

Step 2: Key Formula or Approach:
In a right-angled triangle containing reference angle \(A\):
\[ \sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} \] \[ \cot A = \frac{\text{Adjacent}}{\text{Opposite}} \] We can apply the Pythagoras theorem to calculate the missing adjacent side:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \] Once the adjacent side is determined, we can evaluate the ratio for \(\cot A\).

Step 3: Detailed Explanation:

• Let us assume the opposite side is \(1k\) and the hypotenuse is \(3k\), where \(k\) is a positive scale factor.
For ease of calculation, we can set \(k = 1\):
\(\text{Opposite} = 1\)
\(\text{Hypotenuse} = 3\)

• Apply the Pythagoras theorem to find the length of the adjacent side:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \] \[ 3^2 = 1^2 + \text{Adjacent}^2 \] \[ 9 = 1 + \text{Adjacent}^2 \] \[ \text{Adjacent}^2 = 9 - 1 \] \[ \text{Adjacent}^2 = 8 \]

• Take the square root to find the length of the adjacent side:
\[ \text{Adjacent} = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \]

• Substitute the lengths of the adjacent and opposite sides into the formula for \(\cot A\):
\[ \cot A = \frac{\text{Adjacent}}{\text{Opposite}} \] \[ \cot A = \frac{2\sqrt{2}}{1} \] \[ \cot A = 2\sqrt{2} \]

Step 4: Final Answer:
The value of \(\cot A\) is \(2\sqrt{2}\).
Therefore, the correct option is (B).