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

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Answer 1 · 2018-03-05T22:12:06

III

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.