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

1 Answer

Answer:

#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#