In the given figure, the measure of angle 𝐵𝐴𝐶 is equal to 90 degrees, where the line 𝐴𝐷 is perpendicular to 𝐵𝐶. What is the value of 𝐴𝐵 cos 𝜃?
So the first thing we can do is add this angle. So it’s the angle 𝐴𝐷𝐵. We know that it’s a right angle, 90 degrees. And we know that because we’re told that the line 𝐴𝐷 is perpendicular, so therefore at right angles too the line 𝐵𝐶. So now, we have a right-angle triangle. And that triangle 𝐴𝐵𝐷 is useful because it includes the line 𝐴𝐵 that we’re looking for.
Well, the first thing we’re gonna do is label our triangle. So the first label I’m gonna put on is the hypotenuse because this is the longest side opposite the right angle. And then, we have the opposite. And that’s because this is the side opposite the angle 𝜃. And then finally adjacent because this is the side that’s next to the angle 𝜃, but also between 𝜃 and the right angle.
So now, the next thing we’re gonna do is look at the trigonometric ratios. And we can do that because, as we said, we’re looking at a right-angle triangle. So to help us look at these, we’ve got our memory aid, which is SOHCAHTOA. And if we take a look at the question, we can see that we’re looking for 𝐴𝐵 cos 𝜃. So therefore, we’re gonna be interested in CAH, the part that deals with the cosine ratio. And what this tells us is that the cos of 𝜃 is equal to the adjacent divided by the hypotenuse.
So therefore, if we take a look at our diagram, we’ve got the adjacent and the hypotenuse. And I’ve circled these. And they are 𝐴𝐷 and 𝐴𝐵, respectively. So therefore, we can say that the cos of 𝜃 is going to be 𝐴𝐷 over 𝐴𝐵. And that’s because that’s the adjacent divided by the hypotenuse.
So now what we can do is multiply both sides of our equation by 𝐴𝐵. And that’s because, first of all, it removes it from being the denominator removes the fractional element of our equation. But also, it will give us our 𝐴𝐵 cos 𝜃 that we’re looking for. So therefore, this would give us that 𝐴𝐵 cos 𝜃 is equal to 𝐴𝐷.
So therefore, we can say that the value of 𝐴𝐵 cos 𝜃 is going to be equal to 𝐴𝐷.