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Understanding the signs of trigonometric functions such as sine, cosine, and tangent in different quadrants of the coordinate system is pivotal for grasping the concept of angle measurement in trigonometry. These signs are determined based on the position of an angle's terminal side within a coordinate plane.

In Quadrant I, all trigonometric functions are positive. While in Quadrant II, sine is positive, cosine and tangent are negative. Moving to Quadrant III, tangent becomes positive, and sine and cosine are negative. Lastly, in Quadrant IV, cosine is positive, and sine and tangent are negative.

This pattern can be memorized with the aid of a mnemonic such as 'All Students Take Calculus', where each word represents a quadrant (Starting from Quadrant I and going in counter-clockwise direction) and tells which of the functions are positive in that quadrant.

The coordinate system used in trigonometry is divided into four parts called quadrants, each containing a range of angles with distinct properties for their trigonometric ratios. These quadrants are labeled counterclockwise from the positive x-axis as Quadrant I (0° to 90°), Quadrant II (90° to 180°), Quadrant III (180° to 270°), and Quadrant IV (270° to 360°).

In each quadrant, the signs of the x (cosine) and y (sine) components of an angle are key to determining the sign of its trigonometric functions. Knowing the signs that correspond to each quadrant allows us to pinpoint the location of an angle based on the given trigonometric function's value. In the context of the exercise, \(\cos \theta>0\) indicates that we're either in Quadrant I or IV, and the negative sign of sine function \(\sin \theta<0\) restricts us to Quadrant IV, where the x-values are positive, and y-values are negative.

Sine and cosine are fundamental trigonometric functions that relate the angles of a triangle to the ratios of its sides, particularly in a right-angled triangle.

The sine function (sin), represents the ratio of the opposite side to the hypotenuse. On the unit circle, sine corresponds to the y-coordinate of a point where the terminal side of an angle intersects the circle.

The cosine function (cos), on the other hand, represents the ratio of the adjacent side to the hypotenuse, and on the unit circle, it's the x-coordinate.

One of the key properties of these functions is their periodicity – sine and cosine have a period of \(2\pi\) radians (or 360 degrees) which means their values repeat after every \(2\pi\) radians. Additionally, their values range between -1 and 1. Knowing these properties aids in understanding the behavior of sine and cosine across different quadrants.