Video: Understanding the Cosine Ratio

Bethani Gasparine

In the given figure, 𝑚∠𝐵𝐴𝐶 = 90° and 𝐴𝐷 ⊥ 𝐵𝐶. What is 𝐵𝐶 cos 𝜃?

03:34

Video Transcript

In the given figure, the measure of angle 𝐵𝐴𝐶 is equal to 90 degrees and 𝐴𝐷 is perpendicular to 𝐵𝐶. What is 𝐵𝐶 times the cos of 𝜃?

We’re given a few pieces of information. First, we’re told the measure of angle 𝐵𝐴𝐶 is equal to 90 degrees. And then we’re also told that 𝐴𝐷 is perpendicular to, and perpendicular means they will make a 90-degree angle together. So 𝐴𝐷 is perpendicular to 𝐵𝐶. So this would be a 90-degree angle, and so would this.

So we’re asked to find 𝐵𝐶 times the cos of 𝜃. 𝐵𝐶 is this side length. There isn’t much to do with it. So let’s look at the cos of 𝜃. So here is 𝜃. 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; it’s called the hypotenuse, the side here and here. And we will label these based on the angle that we’re referencing, 𝜃. So if 𝜃 is here, we are referencing this angle. So across from this angle will be the side called opposite. And the side next to it — that’s not a hypotenuse — it’s called the adjacent.

Now if 𝜃 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 𝜃 would be the opposite side divided by the hypotenuse side. Cos of 𝜃 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, let’s look at the cos of 𝜃. So here is angle 𝜃. Now it needs to be in a right triangle. So if we look at the large triangle 𝐵𝐴𝐶, side 𝐴𝐵 would be opposite of 𝜃, 𝐵𝐶 would be the hypotenuse because it’s across from a 90-degree angle at 𝐴, and the adjacent side would be 𝐴𝐶.

So if we want the cos of 𝜃, cosine is adjacent divided by the hypotenuse, so 𝐴𝐶 divided by 𝐵𝐶. So we can replace the cos of 𝜃 with 𝐴𝐶 divided by 𝐵𝐶. The first 𝐵𝐶 we can write over one. And what happens is that 𝐵𝐶s cancel, and we’re left with 𝐴𝐶. So 𝐴𝐶 would be our final answer.

However, what if we looked at triangle 𝐴𝐷𝐶? The cos of 𝜃 would now change because the opposite side would be 𝐴𝐷, the adjacent side would be 𝐶𝐷, and the hypotenuse would be 𝐴𝐶. So adjacent divided by hypotenuse would make the cos of 𝜃 𝐶𝐷 divided by 𝐴𝐶. And if we take 𝐵𝐶 times this, we don’t end up with one segment. So using a larger triangle would be the correct route. So once again, our answer will be 𝐴𝐶.

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