If `sec theta = (25)/(24)`, then what is the value of `sin theta`?

Options

A. `(24)/(25)`
B. `(7)/(25)`
C. `(24)/(7)`
D. `(25)/(7)`

Answer

To solve the problem where \( \sec \theta = \frac{25}{24} \) and we need to find the value of \( \sin \theta \), we can follow these steps:

### Step 1: Understand the relationship of secant
We know that:
\[ \sec \theta = \frac{1}{\cos \theta} \]
So, if \( \sec \theta = \frac{25}{24} \), then:
\[ \cos \theta = \frac{24}{25} \]

### Step 2: Use the Pythagorean identity
We can use the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting \( \cos \theta \):
\[ \sin^2 \theta + \left(\frac{24}{25}\right)^2 = 1 \]

### Step 3: Calculate \( \cos^2 \theta \)
Calculating \( \left(\frac{24}{25}\right)^2 \):
\[ \left(\frac{24}{25}\right)^2 = \frac{576}{625} \]

### Step 4: Substitute and solve for \( \sin^2 \theta \)
Now, substituting back into the Pythagorean identity:
\[ \sin^2 \theta + \frac{576}{625} = 1 \]
Subtract \( \frac{576}{625} \) from both sides:
\[ \sin^2 \theta = 1 - \frac{576}{625} \]
To perform the subtraction, convert 1 to a fraction with the same denominator:
\[ 1 = \frac{625}{625} \]
Thus:
\[ \sin^2 \theta = \frac{625}{625} - \frac{576}{625} = \frac{49}{625} \]

### Step 5: Take the square root to find \( \sin \theta \)
Now, taking the square root of both sides:
\[ \sin \theta = \sqrt{\frac{49}{625}} = \frac{7}{25} \]

### Conclusion
Thus, the value of \( \sin \theta \) is:
\[ \sin \theta = \frac{7}{25} \]