If `sin theta =15/17` then what is the value of `sec theta`

Options

A. `17/8`
B. `8/17`
C. `8/15`
D. `17/15`

Answer

To find the value of \( \sec \theta \) given that \( \sin \theta = \frac{15}{17} \), we can follow these steps:

### Step 1: Understand the relationship of sine
We know that:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{p}{h} \]
Given \( \sin \theta = \frac{15}{17} \), we can identify:
- \( p = 15 \) (opposite side)
- \( h = 17 \) (hypotenuse)

### Step 2: Use the Pythagorean theorem
To find the adjacent side \( b \), we use the Pythagorean theorem, which states:
\[ h^2 = p^2 + b^2 \]
Substituting the known values:
\[ 17^2 = 15^2 + b^2 \]
Calculating \( 17^2 \) and \( 15^2 \):
\[ 289 = 225 + b^2 \]
Now, isolate \( b^2 \):
\[ b^2 = 289 - 225 \]
\[ b^2 = 64 \]
Taking the square root gives:
\[ b = \sqrt{64} = 8 \]

### Step 3: Calculate sec theta
The secant function is defined as:
\[ \sec \theta = \frac{h}{b} \]
Substituting the values we found:
\[ \sec \theta = \frac{17}{8} \]

### Conclusion
Thus, the value of \( \sec \theta \) is:
\[ \sec \theta = \frac{17}{8} \]