Question

How do you find the quadrants in which the terminal side of `theta` must lie given `sintheta` is negative and `tan theta` is positive?

Answer 1

III

Explanation:

We know:

`sintheta="opposite"/"hypotenuse"`

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 `sintheta` is negative, then the opposite side must be negative.

`tantheta= "opposite"/"adjacent"`

If this is positive then the adjacent side must be negative.

i.e

`tantheta=(-"opposite")/-"adjacent"="opposite"/"adjacent"`

So we have both negative x values and negative y values. This must therefore be in the III quadrant.