If `sintheta=(7)/(25)`,then find the value of `tan^(2)theta`.

Options

A. `(7)/(24)`
B. `(49)/(576)`
C. `(1)/(2)`
D. `(49)/(625)`

Answer

To solve the problem where \( \sin \theta = \frac{7}{25} \) and we need to find the value of \( \tan^2 \theta \), we can follow these steps:

### Step 1: Identify the components of the triangle
Given that \( \sin \theta = \frac{7}{25} \), we can interpret this in terms of a right triangle where:
- The opposite side (perpendicular) = 7
- The hypotenuse = 25

### Step 2: Use Pythagorean theorem to find the base
According to the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{perpendicular}^2 + \text{base}^2 \]
Substituting the known values:
\[ 25^2 = 7^2 + \text{base}^2 \]
Calculating the squares:
\[ 625 = 49 + \text{base}^2 \]
Now, isolate \( \text{base}^2 \):
\[ \text{base}^2 = 625 - 49 = 576 \]

### Step 3: Calculate the base
Taking the square root of both sides:
\[ \text{base} = \sqrt{576} = 24 \]

### Step 4: Find \( \tan \theta \)
Now that we have the base, we can find \( \tan \theta \):
\[ \tan \theta = \frac{\text{perpendicular}}{\text{base}} = \frac{7}{24} \]

### Step 5: Calculate \( \tan^2 \theta \)
Now, we need to find \( \tan^2 \theta \):
\[ \tan^2 \theta = \left( \frac{7}{24} \right)^2 = \frac{49}{576} \]

### Final Answer
Thus, the value of \( \tan^2 \theta \) is:
\[ \frac{49}{576} \]

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