Once trigonometric functions are defined through the coordinates of a point P(x, y) on the terminal ray (cosθ = x, sinθ = y, tanθ = y/x, on the unit circle), their signs in each quadrant follow purely from the signs of x and y there — no new formula is needed. In quadrant I both coordinates are positive, so all six functions are positive. In quadrant II, x is negative and y positive, so only sinθ (and its reciprocal cosecθ) stay positive; cosθ, secθ, tanθ, cotθ are negative. In quadrant III both coordinates are negative, so only tanθ (and cotθ, a ratio of two negatives) is positive. In quadrant IV, x is positive and y negative, so only cosθ (and secθ) is positive. This is often remembered by the mnemonic All–Sin–Tan–Cos for quadrants I–II–III–IV respectively. A practical use is finding the sign of a function of a large or negative angle: reduce it to a coterminal angle between 0° and 360° by adding or subtracting whole multiples of 360°, identify that angle's quadrant, and read the sign off the rule above — the function's value is unchanged by this reduction since coterminal angles give the same point P.