If `"tan" theta=-(4)/(5)`, then `"sin" theta` is:

Options

A. `-(4)/(5)` but not `(4)/(5)`
B. `-(4)/(5)` or `(4)/(5)`
C. `(4)/(5)` but not `-(4)/(5)`
D. None of these

Answer

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

### Step 1: Identify the Quadrant
Since \(\tan \theta\) is negative, \(\theta\) must be in either the second or the fourth quadrant. In the second quadrant, \(\sin \theta\) is positive, while in the fourth quadrant, \(\sin \theta\) is negative.

### Step 2: Set Up the Right Triangle
Using the definition of tangent, we can set up a right triangle where:
- The opposite side (perpendicular) is 4 (since \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = -\frac{4}{5}\)).
- The adjacent side (base) is 5.

### Step 3: Calculate the Hypotenuse
Using the Pythagorean theorem:
\[ \text{Hypotenuse} (H) = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} \]

### Step 4: Calculate \(\sin \theta\)
Now, we can find \(\sin \theta\):
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{\sqrt{41}} \]

### Step 5: Determine the Sign of \(\sin \theta\)
Since \(\tan \theta\) is negative and we are in the second quadrant, \(\sin \theta\) will be positive:
\[ \sin \theta = \frac{4}{\sqrt{41}} \]

### Final Answer
Thus, the value of \(\sin \theta\) is:
\[ \sin \theta = \frac{4}{\sqrt{41}} \]