If `tantheta=(24)/(7)` and `sintheta` is negative then value of `costheta` will be

Options

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

Answer

To solve the problem, we need to find the value of \( \cos \theta \) given that \( \tan \theta = \frac{24}{7} \) and \( \sin \theta \) is negative.

### Step 1: Understand the Quadrant
Since \( \tan \theta \) is positive and \( \sin \theta \) is negative, we can determine the quadrant in which angle \( \theta \) lies.
- \( \tan \theta \) is positive in the 1st and 3rd quadrants.
- \( \sin \theta \) is negative in the 3rd and 4th quadrants.

Thus, \( \theta \) must be in the **3rd quadrant**.

### Step 2: Use the Definition of Tangent
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Given \( \tan \theta = \frac{24}{7} \), we can express this as:
\[ \frac{\sin \theta}{\cos \theta} = \frac{24}{7} \]
This implies:
\[ \sin \theta = 24k \quad \text{and} \quad \cos \theta = 7k \]
for some positive constant \( k \).

### Step 3: Find the Hypotenuse
Using the Pythagorean theorem:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Substituting the expressions for \( \sin \theta \) and \( \cos \theta \):
\[ (24k)^2 + (7k)^2 = 1 \]
Calculating this gives:
\[ 576k^2 + 49k^2 = 1 \]
\[ 625k^2 = 1 \]
\[ k^2 = \frac{1}{625} \]
\[ k = \frac{1}{25} \]

### Step 4: Calculate \( \sin \theta \) and \( \cos \theta \)
Now substituting back to find \( \sin \theta \) and \( \cos \theta \):
\[ \sin \theta = 24k = 24 \times \frac{1}{25} = \frac{24}{25} \]
\[ \cos \theta = 7k = 7 \times \frac{1}{25} = \frac{7}{25} \]

### Step 5: Determine the Sign of \( \cos \theta \)
Since \( \theta \) is in the 3rd quadrant, where cosine is negative, we have:
\[ \cos \theta = -\frac{7}{25} \]

### Final Answer
Thus, the value of \( \cos \theta \) is:
\[ \cos \theta = -\frac{7}{25} \]