Video: Using Trigonometric Ratios to Find Lengths in Right-Angled Triangles

Find the length of 𝐵𝐶 giving the answer to two decimal places.

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Video Transcript

Find the length of 𝐵𝐶, giving the answer to two decimal places.

Before starting to solve this problem, generally what we want to do is to check what strands of mathematics we actually want to use and solve it. So first of all, it has a right angle. So we think, “Okay, what could it be?” So we’re gonna use trigonometry or is it gonna be the Pythagorean theorem? Which one are we going to use?

Well we then know that we’ve actually got a side and we’ve also got an angle. And we’re wanting to find another side. Well therefore, we can say, “Well actually, we know we can use trigonometric ratios because if it was going to be the Pythagorean theorem, we’d have to have at least two known sides, and actually here we’re also given an angle which gives it away.”

Okay, great! So we can get on and solve the problem. Now when we solve the problem like this, I like to break it down into steps. And that’s so that we can solve that logically and actually not miss anything out or make any mistakes. So step one, we’re going to label the sides. So the first side I’m gonna label is the hypotenuse. And this is the hypotenuse because, first of all, it’s the longest side, but also because it’s opposite the right angle. Okay, so that’s the hypotenuse.

The next side I like to label is the opposite. And it’s the opposite because it’s opposite the angle that we’ve been given or the angle that we’re trying to find. And therefore, the final side, which is 𝐵𝐶 or 𝐶𝐵, is going to be 𝐴, which our is adjacent. And it’s our adjacent because it’s next the angle that we’ve been given. Okay, great! So now we completed step one and we’ve labelled the sides. Now for step two, what we need to do is actually choose the ratio, so which ratio we’re going to use.

Now to choose the ratio, what I have is I actually use this mnemonic to help me, which is SOHCAHTOA. You might have other ways to remember your trig ratios, but this is what we’re going to use today. So first of all, we need to identify which is the side that we have and which is the side that we’re looking to find. Well we know 𝐴𝐵 because that’s our opposite and that’s six centimetres. So we know our opposite. And we want to find 𝐵𝐶. So in this case, that’s going to be our adjacent.

Okay, great! So we now have identified these two sides that we have. So therefore, we go to our mnemonic. So we look at SOHCAHTOA. And we actually see well in which part do we actually have both the opposite and the adjacent. Obviously, it’s the final part. So we can say it’s TOA. So therefore, we know that the ratio we’re going to be using is tan. So to remind ourselves, what does TOA mean? Well it actually means that tan of 𝜃 is equal to the opposite divided by the adjacent. And it’d be the simile for the other two. So the sine would be the opposite over the hypotenuse. The cosine of the adjacent divided by hypotenuse, et cetera.

Okay, great! So we’ve now got the ratio that we need to use. So we completed step two. So now for step three, what we’re going to do is we’re actually going to substitute in our values into our trig ratio. So therefore, we can say that tan of 44 degrees is equal to six divided by 𝐵𝐶. Okay, step three complete. And now finally, we’re on to step four, which is our final step. We’re gonna rearrange and we’re gonna solve. And we’re gonna solve this time to find out 𝐵𝐶. So first of all, we’re gonna multiply both sides by the length 𝐵𝐶.

So we’re gonna get 𝐵𝐶 tan 44 degrees is equal to six. And then next, we’re gonna divide both sides by tan 44 degrees. So therefore, we get 𝐵𝐶 is equal to six over tan 44 degrees. This gives us the answer that 𝐵𝐶 is equal to 6.213181883 centimetres. And this point I’m just gonna give you a little tip. And that tip is: make sure you calculate using degrees to make sure there’s a little “deg” or “d” in the display of your calculator cause otherwise you’ll find you’ve got wrong answers. That might be the case if you haven’t got his answer at this point.

Okay then, finally to solve the problem, it wants the answer to two decimal places. So therefore, we can say that our final answer is that the length of 𝐵𝐶, to two decimal places, is 6.21 centimetres.

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