Question Video: Understanding the Tangent Ratio Mathematics • 11th Grade

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


Video Transcript

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

So in this question, we’ve got a right-angle triangle cause we’re told that the measure of angle 𝐵𝐴𝐶 is equal to 90 degrees. So that’s a right angle. So therefore, because we’ve got a right-angle triangle, we know that we can have a look at the trigonometric ratios. And when we’re dealing with the trigonometric ratios, we have a memory aid, which is SOHCAHTOA. And what this does is it tells us how to work out sin 𝜃, cos 𝜃, and tan of 𝜃.

In this question, we’re looking for 𝐴𝐶 tan 𝜃. Well therefore, we know that we’re gonna be interested in TOA because that’s the one that deals with the tangent ratio. And what TOA tells us that we said is that tan 𝜃 is equal to the opposite divided by the adjacent. So what do we do now? Well, the next step is to label our triangle.

Well, the first side that we’re gonna label is our hypotenuse. And that’s because this is the side that’s opposite the right angle. And it’s also the longest side of our triangle. And the triangle we’re interested in is the triangle 𝐴𝐵𝐶. Now, the next side we’re gonna label is the opposite because this is the side opposite the angle, which is 𝜃. And the final side is the adjacent. And this is the side that’s next to the angle 𝜃 and also between the angle 𝜃 and the right angle.

So in this question, we’re interested in the opposite and the adjacent because we’re dealing with tan 𝜃. So therefore, if we substitute these into our formula for tan 𝜃, we can see that tan 𝜃 is equal to 𝐴𝐵, the opposite, divided by 𝐴𝐶, which is the adjacent. And then, what we can do is multiply each side of the equation by 𝐴𝐶 to remove it from the denominator so to get rid of the fraction.

And when we do that, we get 𝐴𝐶 tan 𝜃 is equal to 𝐴𝐵. So that solved the problem because what we were looking for is what 𝐴𝐶 tan 𝜃 was. And 𝐴𝐶 tan 𝜃, as we’ve already said, is 𝐴𝐵.

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