𝐴𝐵𝐶 is a right triangle at
𝐵. Find the cot of 𝛼 given that the
cot of 𝜃 is four-thirds.
Because we know that 𝐴𝐵𝐶 is a
right triangle, we can also say that 𝐴𝐵𝐷 is a right triangle. And this means we can identify
angle 𝐴𝐷𝐵 as 90 degrees minus 𝜃. It also means we can say that 𝛼
plus 90 degrees minus 𝜃 is equal to 180 degrees as we know that 𝐵𝐶 forms a
straight line. And then if we subtract 90 degrees
from both sides of this equation, we see that 90 degrees equals 𝛼 minus 𝜃. And adding 𝜃 to both sides tells
us that 𝛼 equals 90 degrees plus 𝜃, which means the cot of 𝛼 is equal to the cot
of 90 degrees plus 𝜃. And now it seems like we’re getting
closer because we know cot in terms of 𝜃. Based on our cofunction identity,
we know that the tan of 90 degrees minus 𝜃 equals the cot of 𝜃, so we want to
rearrange the cot of 90 degrees plus 𝜃.
We can rewrite it to be the cot of
90 degrees minus negative 𝜃. And then, we’ll rewrite this in
terms of tangent because the cotangent is the reciprocal of tangent. We can say that this is equal to
one over tan of 90 degrees minus negative 𝜃 and that tan of 90 degrees minus
negative 𝜃 simplifies to the cot of negative 𝜃. But now we have one over cot of
negative 𝜃, which will be equal to the tan of negative 𝜃. And since tan of negative 𝜃 is
equal to the negative tan of 𝜃, it’s an odd function, which means we simplify to
the negative tan of 𝜃.
Going back to our diagram, if the
cot of 𝜃 is four-thirds, 𝐴𝐵 equals four, 𝐵𝐷 equals three, the tan of 𝜃 is the
opposite over the adjacent side length, which here is three-fourths. And we need the negative tangent,
which will be negative three-fourths. We’ve shown that the cot of 𝛼 will
be equal to the negative tan of 𝜃 and it’s negative three-fourths.