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

Options

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

Answer

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

### Step 1: Understand the definition of cotangent
The cotangent of an angle in a right triangle is defined as the ratio of the adjacent side (base) to the opposite side (perpendicular). Thus, if \( \cot \theta = \frac{7}{8} \), we can identify:
- Adjacent side (base) = 7
- Opposite side (perpendicular) = 8

### Step 2: Use the Pythagorean theorem to find the hypotenuse
According to the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2 \]
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 the value of \( \sin \theta \)
The sine of an angle is defined as the ratio of the opposite side (perpendicular) to the hypotenuse. Therefore:
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{8}{\sqrt{113}} \]

### Step 4: Rationalize the denominator (optional)
To express \( \sin \theta \) in a more standard form, we can rationalize the denominator:
\[ \sin \theta = \frac{8}{\sqrt{113}} \cdot \frac{\sqrt{113}}{\sqrt{113}} = \frac{8\sqrt{113}}{113} \]

### Final Answer
Thus, the value of \( \sin \theta \) is:
\[ \sin \theta = \frac{8}{\sqrt{113}} \quad \text{or} \quad \sin \theta = \frac{8\sqrt{113}}{113} \]

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