Why is the sine of quadrant-four angles negative? If both the hypotenuse and the opposite lengths are negative, and neg/neg is positive so why are calculators giving a negative value?
After reading some answers, I understand a sign is not supposed to be added to the hypotenuse, but why is it then allowed for the length opposite to the angle in question?
 A: *

*A length—including a triangle hypotenuse—is always nonnegative.
(It doesn't make sense to say that a triangle side has length $-7.)$


*In general (i.e., when not dealing with acute angles), think of trigonometric ratios not as ratios of lengths, but as ratios of $x$- and $y$- coordinates on the unit circle (as illustrated in Golden_Ratio's answer).
If this is too abstract, then think of them as your usual ratios of triangle sides, except that the non-hypotenuse sides now have signed lengths (each signed according to its location in the Cartesian coordinate plane).

why is a sign then allowed for the opposite lengths of the angle in question?

This is merely by definition/convention (trigonometric ratios for acute angles also start from our <opposite/hypotenuse, etc.> definitions of them): the $\pm$ sign of each non-hypotenuse side is assigned according to the horizontal/vertical displacement (rather than distance) of the corresponding point on the circle from the origin.
A: $\sin$ and $\cos$ return the sine and cosine of the included angle, which aren't necessarily the $\sin$ and $\cos$ values of the required angle.
Use the parity of the $\sin$ and $\cos$ functions to determine the actual value.
A: A quadrant 3 angle has the property that the $x$ and $y$ values are both negative. A quadrant 4 angle, on the other hand, has a positive $x$-value.
If you are looking at the Cartesian plane with the $x$-axis running horizontally, and the $y$-axis running vertically, quadrant $1$ is in the top right, quadrant $2$ the top left, quadrant $3$ the bottom left, and quadrant $4$ is in the bottom right.
A: The geometric interpretation of cosine and sine of an angle are the x coordinate and y coordinate respectively associated with the angle on the unit circle. Tangent is the ratio sine/cosine. An angle in quadrant 4 thus has positive cosine and negative sine. Here is the more general picture:

