If `sintheta=(4)/(5)`, then the value of `sectheta.tantheta` is

Options

A. `-9//20`
B. `9//20`
C. `-20//9`
D. `20//9`

Answer

To find the value of \( \sec \theta \cdot \tan \theta \) given that \( \sin \theta = \frac{4}{5} \), we can follow these steps:

### Step 1: Understand the relationship between sine, cosine, and tangent.
We know that:
\[ \sin \theta = \frac{\text{perpendicular}}{\text{hypotenuse}} \]
From the given \( \sin \theta = \frac{4}{5} \), we can identify:
- Perpendicular (opposite side) = 4
- Hypotenuse = 5

### Step 2: Use the Pythagorean theorem to find the base (adjacent side).
Using the Pythagorean theorem:
\[ \text{hypotenuse}^2 = \text{perpendicular}^2 + \text{base}^2 \]
Substituting the known values:
\[ 5^2 = 4^2 + \text{base}^2 \]
\[ 25 = 16 + \text{base}^2 \]
\[ \text{base}^2 = 25 - 16 = 9 \]
\[ \text{base} = 3 \]

### Step 3: Calculate \( \cos \theta \).
Using the definition of cosine:
\[ \cos \theta = \frac{\text{base}}{\text{hypotenuse}} = \frac{3}{5} \]

### Step 4: Calculate \( \sec \theta \).
The secant function is the reciprocal of cosine:
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{3}{5}} = \frac{5}{3} \]

### Step 5: Calculate \( \tan \theta \).
Using the definition of tangent:
\[ \tan \theta = \frac{\text{perpendicular}}{\text{base}} = \frac{4}{3} \]

### Step 6: Find \( \sec \theta \cdot \tan \theta \).
Now we can find the product:
\[ \sec \theta \cdot \tan \theta = \left(\frac{5}{3}\right) \cdot \left(\frac{4}{3}\right) = \frac{20}{9} \]

### Final Answer:
Thus, the value of \( \sec \theta \cdot \tan \theta \) is:
\[ \frac{20}{9} \]

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