Video: Using Right-Angled Triangle Trigonometry to Find Lengths in Word Problems

A ladder is leaning against a wall where the upper end is 4 m from the ground. The angle of inclination of the ladder that to the ground is 40°. Find the horizontal distance between the base of the ladder and the wall giving the answer to two decimal places.

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Video Transcript

A ladder is leaning against a wall, where the upper end is four meters from the ground. The angle of inclination of the ladder that to the ground is 40 degrees. Find the horizontal distance between the base of the ladder and the wall, giving the answer to two decimal places.

First of all, to help us solve this problem I have drawn us a little sketch and I’ve actually drawn the ladder here leaning against the wall. Now we’re gonna start adding some information to our diagram. And we’ve actually got the first bit of information, which is that the upper end is four meters from the ground. So therefore, we actually know that the vertical height is four meters. And I’ve marked this on our diagram.

Next, we know that the angle inclination of the ladder to the ground is 40 degrees. So I’ve marked this on our diagram. And then finally, we have what we want to find, which is the horizontal distance between the base of the ladder and the wall. So I’ve marked this on as 𝑥. In order to be able to solve this problem, what we actually want to do is we actually want to model this as a right- angled triangle so I have included that there because we’ve got the horizontal distance is what we want to know and we know the vertical distance. So therefore, we now formed a right- angled triangle. So we can actually use this to solve the problem.

I’m gonna go through the problem in steps. But before we even do, what we need to actually decide is what we’re going to use to help us solve it. We’ve seen that it’s a right-angled triangle. So we know that we can either use the trigonometric ratios, or trig ratios, or the Pythagorean theorem. Well in this question, we’ve actually got a side that we know and side we want to find out, and an angle. So therefore, we’re gonna use the trig ratios. And that’s because if we’re going to use Pythagorean theorem, we’d want two sides, maybe a third, that we wanted to find out.

Okay, so great! Let’s get on with step one. Well step one is to actually label the sides on our diagram. The first side that we’re gonna label is the hypotenuse, because a hypotenuse is always opposite the right angle and it’s the longest side. The next side I always like to label is the opposite. And that’s because it’s opposite the angle that we’ve got or an angle that we’re trying to find. And then finally, I label the adjacent. That’s the one that’s next the angle that we have or the angle we’re trying to find. So great! We’ve got all of our sides labelled and we completed step one.

Now we’re gonna move on to step two. And for step two, we need to choose the ratio. And what I mean by this is we want to choose our trigonometric ratio. And to enable me to do this, what I like to use is this mnemonic which is SOHCAHTOA. Okay, so how am I gonna use this to help me find which ratio it is? First of all, I actually circle the sides that we either have or are looking for. So in this case, we have the opposite cause that’s four meters. And we want to find the adjacent, which I’ve called 𝑥.

Okay, so we have 𝑂 and 𝐴. And then next, what I do is I go back to our mnemonic. And I have a look at, okay, which part has the 𝑂 and the 𝐴, the opposite and the adjacent. And here we go. I found it! It’s TOA. So therefore, we know that we’re going to use the trig ratio that tan 𝜃 is equal to the opposite divided by the adjacent. Okay, great! Step two complete. And now we’re gonna move on to step three. And for step three, what we’re gonna do is we’re actually we gonna substitute our values in. So we’re gonna substitute our values in to tan 𝜃 is equal to the opposite over the adjacent.

So when we do that, we get that tan of 40 degrees is equal to four over 𝑥. So therefore, we can move on to our final step, step four, where we gonna rearrange and solve. So therefore, we’re gonna do first stage, which is multiply both sides by 𝑥, which is gonna give us 𝑥 tan 40 degrees equals four. Then our next stage is to divide both sides by tan of 40 degrees. So we’re left with 𝑥 is equal to four divided by tan 40 degrees. And then when we calculate this, we get 𝑥 is equal 4.76701437.

Okay, so now what we need to do to finish the problem, well the final part is actually to around our answer to two decimal places. So therefore, we get that 𝑥 is equal to 4.77 meters. And that’s like, as we said, to two decimal places. So therefore, answering the question, we can say that the horizontal distance between the base of the ladder and the wall is 4.77 meters.

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