Q.State the quadrant in which θ lies if $\tan\theta < 0$ and $\sec\theta > 0$.

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 …