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

Mar 5, 2018

III

#### Explanation:

We know:

$\sin \theta = \text{opposite"/"hypotenuse}$

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 $\sin \theta$ is negative, then the opposite side must be negative.

$\tan \theta = \text{opposite"/"adjacent}$

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.