If `cos theta=(3)/(5)` then the value of `cot theta` is :

Options

A. `(4)/(5)`
B. `(3)/(4)`
C. `(4)/(3)`
D. None of these

Answer

To find the value of \( \cot \theta \) given that \( \cos \theta = \frac{3}{5} \), we can follow these steps:

### Step 1: Understand the relationship between cosine and the sides of a right triangle.
Given \( \cos \theta = \frac{b}{h} \), where \( b \) is the adjacent side and \( h \) is the hypotenuse. Here, we have:
- \( b = 3 \) (adjacent side)
- \( h = 5 \) (hypotenuse)

### Step 2: Use the Pythagorean theorem to find the opposite side.
According to the Pythagorean theorem:
\[ h^2 = b^2 + p^2 \]
where \( p \) is the opposite side. Plugging in the values:
\[ 5^2 = 3^2 + p^2 \]
\[ 25 = 9 + p^2 \]
Subtract 9 from both sides:
\[ p^2 = 25 - 9 = 16 \]
Taking the square root:
\[ p = \sqrt{16} = 4 \]

### Step 3: Calculate \( \cot \theta \).
The cotangent function is defined as:
\[ \cot \theta = \frac{b}{p} \]
Substituting the values we found:
\[ \cot \theta = \frac{3}{4} \]

### Final Answer:
Thus, the value of \( \cot \theta \) is \( \frac{3}{4} \).

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