How do you find the quadrants in which the terminal side of #theta# must lie given #sintheta# is negative and #tan theta# is positive?
The hypotenuse is always positive by definition:
i.e. The square on the hypotenuse is equal to the sum of the squares of the other two sides. The square of a number is always positive.
If this is positive then the adjacent side must be negative.
So we have both negative x values and negative y values. This must therefore be in the III quadrant.