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We have triangle ABC here, which
looks like a right triangle. And we know it's
a right triangle because 3 squared plus 4
squared is equal to 5 squared. And they want us to figure out
what cosine of 2 times angle ABC is. So that's this angle-- ABC. Well, we can't
immediately evaluate that, but we do know what the
cosine of angle ABC is. We know that the cosine of
angle ABC-- well, cosine is just adjacent
over hypotenuse. It's going to be equal to 3/5. And similarly, we know what
the sine of angle ABC is. That's opposite over hypotenuse. That is 4/5. So if we could break this
down into just cosines of ABC and sines of ABC, then we'll
be able to evaluate it. And lucky for us, we have a
trig identity at our disposal that does exactly that. We know that the cosine
of 2 times an angle is equal to cosine
of that angle squared minus sine of that
angle squared. And we've proved
this in other videos, but this becomes very
helpful for us here. Because now we know
that the cosine-- Let me do this in
a different color. Now, we know that the
cosine of angle ABC is going to be equal
to-- oh, sorry. It's the cosine of 2
times the angle ABC. That's what we care about. 2 times the angle
ABC is going to be equal to the cosine of angle ABC
squared minus sine of the angle ABC squared. And we know what
these things are. This thing right
over here is just going to be equal
to 3/5 squared. Cosine of angle a ABC is 3/5. So we're going to square it. And this right over here
is just 4/5 squared. So it's minus 4/5 squared. And so this simplifies
to 9/25 minus 16/25, which is equal to 7/25. Sorry. It's negative. Got to be careful there. 16 is larger than 9. Negative 7/25. Now, one thing that
might jump at you is, why did I get a
negative value here when I doubled the angle here? Because the cosine was
clearly a positive number. And there you just have to
think of the unit circle-- which we already know the unit circle
definition of trig functions is an extension of the
Sohcahtoa definition. X-axis. Y-axis. Let me draw a unit circle here. My best attempt. So that's our unit circle. So this angle right over here
looks like something like this. And so you see its
x-coordinate-- which is the cosine of that
angle-- looks positive. But then, if you were
to double this angle, it would take you out
someplace like this. And then, you see-- by the
unit circle definition-- the x-coordinate, we are now
sitting in the second quadrant. And the x-coordinate
can be negative. And that's essentially what
happened in this problem.