If `tantheta=(2)/(3)`, then `sintheta`=

Options

A. `+-0.55`
B. `+-0.4`
C. 0.55
D. 0.83

Explanation

To find \( \sin \theta \) given that \( \tan \theta = \frac{2}{3} \), we can follow these steps:

### Step 1: Understand the relationship of tangent in a right triangle
We know that:
\[ \tan \theta = \frac{\text{perpendicular}}{\text{base}} \]
From the problem, we have:
\[ \tan \theta = \frac{2}{3} \]
This means we can consider a right triangle where the perpendicular side (opposite to angle \( \theta \)) is 2 and the base (adjacent to angle \( \theta \)) is 3.

### Step 2: Calculate the hypotenuse
Using the Pythagorean theorem, we can find the hypotenuse \( h \):
\[ h = \sqrt{(\text{perpendicular})^2 + (\text{base})^2} = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 3: Calculate \( \sin \theta \)
The sine of an angle in a right triangle is given by:
\[ \sin \theta = \frac{\text{perpendicular}}{\text{hypotenuse}} = \frac{2}{\sqrt{13}} \]
To express this in a more standard form, we can rationalize the denominator:
\[ \sin \theta = \frac{2}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}} = \frac{2\sqrt{13}}{13} \]

### Step 4: Determine the sign of \( \sin \theta \)
Since \( \tan \theta \) is positive, \( \theta \) could be in either the first or third quadrant. In the first quadrant, both sine and tangent are positive, while in the third quadrant, sine is negative. Therefore, we have:
\[ \sin \theta = \pm \frac{2\sqrt{13}}{13} \]

### Final Answer
Thus, the value of \( \sin \theta \) is:
\[ \sin \theta = \pm \frac{2\sqrt{13}}{13} \]

---