If sin ` alpha = (X)/( sqrt(X ^(2) + Y^(2)))` then what is the value of tan ` alpha `

Options

A. x/y
B. y/x
C. y
D. None of these

Answer

To find the value of \( \tan \alpha \) given that \( \sin \alpha = \frac{X}{\sqrt{X^2 + Y^2}} \), we can follow these steps:

### Step 1: Identify the components of the triangle
From the definition of sine, we know:
\[ \sin \alpha = \frac{\text{Perpendicular}}{\text{Hypotenuse}} \]
In this case, we have:
- Perpendicular = \( X \)
- Hypotenuse = \( \sqrt{X^2 + Y^2} \)

### Step 2: Use the Pythagorean theorem to find the base
According to the Pythagorean theorem, we can find the base (adjacent side) using:
\[ \text{Base}^2 = \text{Hypotenuse}^2 - \text{Perpendicular}^2 \]
Substituting the values we have:
\[ \text{Base}^2 = (\sqrt{X^2 + Y^2})^2 - X^2 \]
This simplifies to:
\[ \text{Base}^2 = (X^2 + Y^2) - X^2 = Y^2 \]
Taking the square root gives us:
\[ \text{Base} = Y \]

### Step 3: Calculate \( \tan \alpha \)
Now that we have both the perpendicular and the base, we can find \( \tan \alpha \):
\[ \tan \alpha = \frac{\text{Perpendicular}}{\text{Base}} = \frac{X}{Y} \]

### Final Answer
Thus, the value of \( \tan \alpha \) is:
\[ \tan \alpha = \frac{X}{Y} \]

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