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In the world of trigonometry, three fundamental functions—sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \))—form the core of understanding angles. Trigonometric ratios are relationships between the angles and side lengths in right triangles.

- **Sine** is defined as the ratio of the length of the side opposite the angle to the hypotenuse of the triangle; i.e., \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \). It reflects the vertical movement and energy in the angle.
- **Cosine** measures how much of the angle fits onto the horizontal plane, defined as the ratio of the adjacent side to the hypotenuse; i.e., \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).
- **Tangent** is the quotient of sine and cosine. It's essentially the slope at that point defined by \( \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\text{opposite}}{\text{adjacent}} \).

These ratios are not just limited to right triangles; they extend to angles of any measure through the unit circle. Variations in their signs reveal key information about their position on the coordinate plane.

The unit circle is an invaluable tool in trigonometry, representing all possible angles from 0 to 360 degrees (or 0 to 2\(\pi\) radians) on a circle with a radius of 1. This makes many calculations and visualizations simpler.

- The circle's radius equals 1, making the hypotenuse of any right triangle formed within the circle always equal to 1.
- Each point on the circle's circumference corresponds to a position where one can measure sine, cosine, and hence, tangent.
- The x-coordinate of a point on this circle represents the cosine of the angle, while the y-coordinate represents the sine of the angle. Hence, the Cartesian coordinates (\( x, y \)) equate to (\( \cos \theta, \sin \theta \)).

The unit circle helps unify the understanding of trigonometric functions with geometric interpretations, aiding in comprehension of angles across different quadrants.

The coordinate plane is divided into four quadrants, each indicating different characteristics of trigonometric functions. Here's a breakdown of what happens in each quadrant:

- **First Quadrant (0° to 90°):** Here, both sine and cosine are positive. Since tangent is the ratio of sine to cosine, it too stays positive.
- **Second Quadrant (90° to 180°):** Sine remains positive while cosine becomes negative. Consequently, tangent, the quotient, also becomes negative. This aligns with conditions like \( \tan \theta < 0 \) and \( \sin \theta > 0 \), such as in the original problem statement.
- **Third Quadrant (180° to 270°):** Both sine and cosine are negative, so their quotient, tangent, is positive. This quadrant is less applicable to the given problem, as it does not satisfy our condition of positive sine.
- **Fourth Quadrant (270° to 360°):** Sine becomes negative and cosine positive, so tangent returns to a negative value. This polarity differs from the second quadrant by the sine sign.

Understanding these nuances in each quadrant helps solve trigonometry problems, discerning which quadrant an angle should fall in based on the trigonometric function values.