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- [Instructor] In triangle
ABC above, AB is equal to BC, so this is equal to that. What is the value of sine of angle CBD minus cosine of angle BAC? Pause this video and see
if you can figure that out. All right, now let's work
through this together. And before I even look at
this expression on the right, let me figure out what else
I know about this triangle. So I know what this angle is. It's going to be 34 degrees. How do I know that? Because if I have an
isosceles triangle like this, where these two sides are equivalent, then the base angles are going to have the same measure as well. We also know that if
this angle is 90 degrees, that this angle has to
be 90 degrees as well, because they have to add up to 180. And then we also know if you
have two angles in common, in a triangle, in two
corresponding triangles, then the third angle is
going to be in common. So we know that this angle is going to be, it's going to have the same
measure as that angle there. And we also know that if you
have three angles like this that all have the same measure
in two different triangles, and you have a side
between two of those angles that is congruent, and you have
two sides of those triangles that are congruent to
each other, and we do, we have this side is
congruent to that side and we have this side is of
course congruent to itself. Well, then that means that the third side is going to be congruent to
the corresponding third side in the other triangle, that these are completely
congruent triangles. Now based on all of that, let's address the elephant
in the room, so to speak. Let's see if we can figure
out this expression. So now let's think
about sine of angle CBD, CBD, we're talking about
this angle right over here, the sine is opposite over hypotenuse. Opposite is DC, hypotenuse is opposite the
90 degree side, so that's BC. And then we are going to
subtract cosine of angle BAC. BAC is this angle right over here. Cosine is adjacent over hypotenuse. If what I'm saying is unfamiliar, I encourage you to review the
right triangle trigonometry, and if the things I did
about segment congruence and congruent triangles and
similar triangles unfamiliar, I encourage you to review
that on Khan Academy. But cosine of angle BAC, that's
adjacent over hypotenuse. Adjacent is AD and then hypotenuse is AB. Now, how do we figure out what
this is going to be equal to? Well, we know a few things already. We know that AB is equal to BC, so we can rewrite this as AB. We also know that AD is equal to DC, so we could write this as AD, and now this is starting
to become quite clear. This is AD over AB minus AD over AB, which is going to be equal
to zero, and we're done.