If `cot alpha = (4)/( 7 . 5)` then what is the value of `sec alpha` and `sin alpha `

Options

A. ` (17)/(8),(17)/(15)`
B. `(17)/(8),(15)/(17)`
C. `(8)/(15),(15)/(17)`
D. None of these

Answer

To solve the problem where cot alpha = 4/7.5, we need to find the values of sec alpha and sin alpha. Let's go through the steps systematically.

### Step 1: Convert cot alpha to a simpler fraction
Given:
\[ \cot \alpha = \frac{4}{7.5} \]

To eliminate the decimal, we can multiply both the numerator and the denominator by 2:
\[ \cot \alpha = \frac{4 \times 2}{7.5 \times 2} = \frac{8}{15} \]

### Step 2: Understand the relationship of cotangent
The cotangent of an angle in a right triangle is defined as:
\[ \cot \alpha = \frac{\text{Base}}{\text{Perpendicular}} \]
From the fraction \( \frac{8}{15} \), we can identify:
- Base = 8
- Perpendicular = 15

### Step 3: Calculate the hypotenuse
Using the Pythagorean theorem:
\[ h = \sqrt{(\text{Base})^2 + (\text{Perpendicular})^2} \]
\[ h = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \]

### Step 4: Calculate sec alpha
The secant of an angle is defined as:
\[ \sec \alpha = \frac{\text{Hypotenuse}}{\text{Base}} \]
Substituting the values we found:
\[ \sec \alpha = \frac{17}{8} \]

### Step 5: Calculate sin alpha
The sine of an angle is defined as:
\[ \sin \alpha = \frac{\text{Perpendicular}}{\text{Hypotenuse}} \]
Substituting the values we found:
\[ \sin \alpha = \frac{15}{17} \]

### Final Answers
Thus, we have:
- \( \sec \alpha = \frac{17}{8} \)
- \( \sin \alpha = \frac{15}{17} \)