If `cot theta =7/8` .Find the value of `cos theta `

Options

A. `8/113`
B. `7/(sqrt(113))`
C. `8/(sqrt(113))`
D. `6/(sqrt(113))`

Answer

To find the value of \( \cos \theta \) given that \( \cot \theta = \frac{7}{8} \), we can follow these steps:

### Step 1: Understand the relationship of cotangent
We know that:
\[ \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{7}{8} \]
This means that in a right triangle, the length of the adjacent side (base) is 7 and the length of the opposite side (perpendicular) is 8.

### Step 2: Use the Pythagorean theorem to find the hypotenuse
According to the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Adjacent}^2 + \text{Opposite}^2 \]
Substituting the values we have:
\[ \text{Hypotenuse}^2 = 7^2 + 8^2 \]
Calculating the squares:
\[ \text{Hypotenuse}^2 = 49 + 64 = 113 \]
Now, take the square root to find the hypotenuse:
\[ \text{Hypotenuse} = \sqrt{113} \]

### Step 3: Find \( \cos \theta \)
We know that:
\[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
Substituting the values we found:
\[ \cos \theta = \frac{7}{\sqrt{113}} \]

### Step 4: Rationalize the denominator (optional)
To rationalize the denominator, we can multiply the numerator and denominator by \( \sqrt{113} \):
\[ \cos \theta = \frac{7 \sqrt{113}}{113} \]

### Final Answer
Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = \frac{7}{\sqrt{113}} \quad \text{or} \quad \frac{7 \sqrt{113}}{113} \]

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