If tantheta=`(-4)/(5)`,then sintheta is

Options

A. `(-4)/(√41)`but not `(4)/√41)`
B. `(-4)/(√41)`or(4)/(√41)`
C. `(4)/(√41)`but not`(-4)/(√41)`
D. None of these

Answer

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

### Step 1: Understand the relationship of tan, sin, and cos
We know that:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Given \( \tan \theta = -\frac{4}{5} \), we can interpret this as:
\[ \frac{\text{Opposite}}{\text{Adjacent}} = -\frac{4}{5} \]
This means that the opposite side (perpendicular) can be -4 and the adjacent side (base) is 5.

### Step 2: Construct a right triangle
We can visualize this as a right triangle where:
- The opposite side is -4 (we will take the absolute value for length)
- The adjacent side is 5

### Step 3: Calculate the hypotenuse
Using the Pythagorean theorem:
\[ \text{Hypotenuse}^2 = \text{Opposite}^2 + \text{Adjacent}^2 \]
Substituting the values:
\[ \text{Hypotenuse}^2 = (-4)^2 + 5^2 = 16 + 25 = 41 \]
Thus, the hypotenuse is:
\[ \text{Hypotenuse} = \sqrt{41} \]

### Step 4: Calculate \( \sin \theta \)
Now, we can find \( \sin \theta \):
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{-4}{\sqrt{41}} \]
Since sine can be positive or negative depending on the quadrant, we take the absolute value of the opposite side for the length:
\[ \sin \theta = \frac{4}{\sqrt{41}} \quad \text{or} \quad \sin \theta = \frac{-4}{\sqrt{41}} \text{ (depending on the quadrant)} \]

### Conclusion
Thus, the value of \( \sin \theta \) is:
\[ \sin \theta = \frac{4}{\sqrt{41}} \quad \text{or} \quad \sin \theta = -\frac{4}{\sqrt{41}} \]