Hence, costheta = -4/5. We know that tantheta>0, and the only quadrant where tangent is positive and the other trig ratios are negative is quadrant III.
Since we only know the side adjacent theta and the hypotenuse, we must find the side opposite theta.
We can do this using Pythagorean theorem. Let a be -4 and c be 5.
a^2 + b^2 = c^2
b^2 = c^2 - a^2
b^2 = (5)^2 - (-4)^2
b ^2= 25 - 16
b = sqrt(9)
b = +-3
We will take the -3, because in quadrant three both opposite and adjacent sides to the angle theta will be negative.
Now that we know that
"adjacent = -4"
"opposite"= -3
"hypotenuse = 5"
We can define cosine and cotangent. Cosine is adjacent/hypotenuse, and cotangent is 1/tantheta = 1/("opposite"/"adjacent") = "adjacent"/"opposite".
Applying these definitions to the problem at hand, we have:
cottheta = 4/3
costheta = -4/5
Now, adding these is simple arithmetic.
4/3 + (-4/5) = 20/15 - 12/15 = 8/15
Thus, costheta + cot theta = 8/15.
Hopefully this helps!
