Question

If `sintheta=3/5`, what is `costheta + cottheta`?

Answer 1

`costheta + cottheta="adj"/"hyp"+"adj"/"opp"=4/5+4/3=12/15+20/15=32/15` in Q1. `-32/15` in Q2.

Explanation:

If `sintheta=3/5`, what is `costheta + cottheta`?

First, let's remember that `sintheta` is the ratio of the opposite side divided by the hypotenuse:

`sin="opp"/"hyp"`

So we know with this triangle that we have a side = 3 and hypotenuse = 5.

There is a special kind of right triangle called the "3, 4, 5 triangle", so called because of the ease in calculating sides using the pythagorean theorem:

`a^2+b^2=c^2`

And so if we didn't know or remember the "3, 4, 5 triangle", we could find it this way:

`3^2+b^2=5^2`

`9+b^2=25`

`b^2=16`

`b=4`

With opposite = 3, adjacent = 4, and hypotenuse = 5, we can answer the question.

Ok, so `sintheta` is positive, which means the opposite is positive (the hypotenuse is always positive). The quadrants on the cartesian chart where the opposite is positive (i.e. has a positive y-value and not a negative one) is in Q1 and Q2. So let's solve for each quadrant:

For Q1:
`costheta + cottheta="adj"/"hyp"+"adj"/"opp"=4/5+4/3=12/15+20/15=32/15`

For Q2:
`costheta + cottheta="adj"/"hyp"+"adj"/"opp"=-4/5-4/3=-12/15-20/15=-32/15`