### Video Transcript

In the given figure, the measure of
angle π΅π΄πΆ is equal to 90 degrees and π΄π· is perpendicular to π΅πΆ. What is π΅πΆ times the cos of
π?

Weβre given a few pieces of
information. First, weβre told the measure of
angle π΅π΄πΆ is equal to 90 degrees. And then weβre also told that π΄π·
is perpendicular to, and perpendicular means they will make a 90-degree angle
together. So π΄π· is perpendicular to
π΅πΆ. So this would be a 90-degree angle,
and so would this.

So weβre asked to find π΅πΆ times
the cos of π. π΅πΆ is this side length. There isnβt much to do with it. So letβs look at the cos of π. So here is π. Now what does cosine mean? Cosine is a part of the
trigonometric identities. And there is sine, cosine, and
tangent, and these all deal with the right triangles. And with right triangles, we need
to label each side.

Across from the 90-degree angle is
the longest side; itβs called the hypotenuse, the side here and here. And we will label these based on
the angle that weβre referencing, π. So if π is here, we are
referencing this angle. So across from this angle will be
the side called opposite. And the side next to it β thatβs
not a hypotenuse β itβs called the adjacent.

Now if π had been in the other
corner, the opposite and the adjacent side would need to switch. But the hypotenuse is always across
from the 90-degree angle. So sine, cosine, and tangent are
all based off of this right triangle. The sine of our angle π would be
the opposite side divided by the hypotenuse side. Cos of π would be the adjacent
side divided by the hypotenuse side. And the tangent of our angle would
be the opposite side divided by the adjacent side.

So once again, letβs look at the
cos of π. So here is angle π. Now it needs to be in a right
triangle. So if we look at the large triangle
π΅π΄πΆ, side π΄π΅ would be opposite of π, π΅πΆ would be the hypotenuse because itβs
across from a 90-degree angle at π΄, and the adjacent side would be π΄πΆ.

So if we want the cos of π, cosine
is adjacent divided by the hypotenuse, so π΄πΆ divided by π΅πΆ. So we can replace the cos of π
with π΄πΆ divided by π΅πΆ. The first π΅πΆ we can write over
one. And what happens is that π΅πΆs
cancel, and weβre left with π΄πΆ. So π΄πΆ would be our final
answer.

However, what if we looked at
triangle π΄π·πΆ? The cos of π would now change
because the opposite side would be π΄π·, the adjacent side would be πΆπ·, and the
hypotenuse would be π΄πΆ. So adjacent divided by hypotenuse
would make the cos of π πΆπ· divided by π΄πΆ. And if we take π΅πΆ times this, we
donβt end up with one segment. So using a larger triangle would be
the correct route. So once again, our answer will be
π΄πΆ.