If `costheta=(35)/(37)` then what is the value of `cottheta` ?

Options

A. `(12)/(35)`
B. `(35)/(12)`
C. `(37)/(12)`
D. `(12)/(37)`

Answer

To find the value of \( \cot \theta \) given that \( \cos \theta = \frac{35}{37} \), we can follow these steps:

### Step 1: Understand the relationship between cosine and cotangent
The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse. The cotangent of an angle is defined as the ratio of the adjacent side to the opposite side. Therefore, we need to find the opposite side to use it in the cotangent formula.

### Step 2: Set up the right triangle
We can visualize a right triangle where:
- The adjacent side (base) is \( 35 \)
- The hypotenuse is \( 37 \)

### Step 3: Use the Pythagorean theorem to find the opposite side
According to the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Adjacent}^2 + \text{Opposite}^2 \]
Substituting the known values:
\[ 37^2 = 35^2 + \text{Opposite}^2 \]
Calculating the squares:
\[ 1369 = 1225 + \text{Opposite}^2 \]
Now, isolate the opposite side:
\[ \text{Opposite}^2 = 1369 - 1225 = 144 \]
Taking the square root:
\[ \text{Opposite} = \sqrt{144} = 12 \]

### Step 4: Calculate \( \cot \theta \)
Now that we have the lengths of the sides, we can find \( \cot \theta \):
\[ \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{35}{12} \]

### Final Answer
Thus, the value of \( \cot \theta \) is \( \frac{35}{12} \).

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