Question Video: Understanding the Cosine Ratio | Nagwa Question Video: Understanding the Cosine Ratio | Nagwa

# Question Video: Understanding the Cosine Ratio Mathematics

In the given figure, πβ π΅π΄πΆ = 90Β° and π΄π· β₯ π΅πΆ. What is π΅πΆ cos π?

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### Video Transcript

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

Weβre given a few pieces of information. First, weβre told the measure of angle π΅π΄πΆ is equal to 90 degrees. And then weβre 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΅πΆ. So this would be a 90-degree angle, and so would this.

So weβre asked to find π΅πΆ times the cos of π. π΅πΆ is this side length. There isnβt much to do with it. So letβs 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; itβs called the hypotenuse, the side here and here. And we will label these based on the angle that weβre 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 thatβs not a hypotenuse β itβs 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, letβs 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΄πΆ, side π΄π΅ would be opposite of π, π΅πΆ would be the hypotenuse because itβs across from a 90-degree angle at π΄, and the adjacent side would be π΄πΆ.

So if we want the cos of π, cosine is adjacent divided by the hypotenuse, so π΄πΆ divided by π΅πΆ. So we can replace the cos of π with π΄πΆ divided by π΅πΆ. The first π΅πΆ we can write over one. And what happens is that π΅πΆs cancel, and weβre left with π΄πΆ. So π΄πΆ would be our final answer.

However, what if we looked at triangle π΄π·πΆ? The cos of π would now change because the opposite side would be π΄π·, the adjacent side would be πΆπ·, and the hypotenuse would be π΄πΆ. So adjacent divided by hypotenuse would make the cos of π πΆπ· divided by π΄πΆ. And if we take π΅πΆ times this, we donβt end up with one segment. So using a larger triangle would be the correct route. So once again, our answer will be π΄πΆ.

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