- The topic discussed is trigonometry and its real-life applications
- The speaker explains how to measure the height of a pole using trigonometry
- They describe forming a triangle with the pole, the distance from the base to the eye, and the line of sight
- The horizontal distance can be easily measured, but the height cannot practically be measured
- Instead, the speaker suggests measuring the angle of the line of sight using a binocular with a protractor
- With the angle and the horizontal distance, the height of the pole can be calculated using trigonometric ratios
- This method can also be applied to measure the height of a mountain or peak

Measuring the height of small objects is easy. But do you know how we can measure the height of really tall objects like mountains? There's a trick, and the name of the trick is trigonometry. Watch the video to find out how we can use trigonometry for measuring the large heights and distances.

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We have a very, very interesting topic at hand today.

1 that has several real life applications.

The topic I'm alluding to is Trigonometry.

Let's see if we can actually measure the height of a peak.

Or for simplicity, instead of the peak, let's start with a pole.

So let's say I have a long pole here and I need to measure its height.

Now for the sake of simplicity, I'm going to assume that I'm actually standing inside a hole of sorts and the ground is at my eye level.

You'll soon realize why this is important.

So the ground is at my eye level, so the base of the pole starts at my eye level and the pole goes on further up from there.

Now we need to measure the height of this pole.

The pole looks like a nice straight line, which is why I picked the pole instead of a mountain.

If you join the pole to my eye, If you join the pole to my eye, that could be the second side of the triangle.

This is nothing but the distance on the ground, how far away I am from the pole.

Now imagine me looking up at the pole and imagine the ray of light that is touching the top end of the pole and reaching my eye.

If you were to imagine that ray of light and you draw a line for it, call it my line of sight, that could be the third side of the triangle, that could be the hypotenuse of the third side of the triangle.

1 side, the vertical side of the triangle is our pole, The horizontal side is the distance between the base of the pole and my eye, and the third side is my line of sight.

All we need to do is measure 1 side of this triangle.

The horizontal distance, if you see, can very easily be measured, right?

I could just start at the base of the pole and walk up to here and measure it.

Let's say we measure it and it's 30 meters.

Now the height of the pole, say the pole is high, we cannot reach the top, you're not gonna do pole climbing, right?

So we cannot practically measure the height of the pole.

If we cannot measure the height of the pole, neither can we measure the distance of the hypotenuse because it's the same logic, correct?

Again, you need to reach the top of the pole to be able to measure the hypotenuse.

So it's not going to be practically possible to measure the hypotenuse.

If we could, then you would simply use Pythagoras' theorem.

If you could measure the hypotenuse, you would use Pythagoras' theorem to find out the height.

If you could measure the hypotenuse, you could very easily measure the height itself.

Now remember we are talking about trigonometry.

Trigonometry is something to do with angles.

1 angle is anyway a right angle triangle, which is why we can apply trigonometry in this triangle.

Can we measure this angle as I look up at this pole, my line of sight?

Can you measure the angle of my line of sight with respect to the horizontal?

Are you ever able to say what is the angle at which you are looking?

Instead of the naked eye, let me take a binocular.

So I can measure the angle at which my binocular is turning.

So I'm looking straight down and I'm going to be able to see the base of the pole.

I slowly move the binocular up as I see more and more of the pole and I keep moving it up until I finally see the top of the pole.

Now if I measure the angle at which the binocular is with respect to the ground, I can easily measure the angle, I already had a protractor fixed here, right?

So I can easily measure the angle, so this angle we now know, right?

Let's say it's 60 degrees, so now we know this angle is 60 degrees and the base, this distance, this length is 30 meters.

Now in this triangle, now in this triangle can we very easily measure the height?

So 1 side is going to be With respect to this angle, if this angle was angle theta, with respect to this angle, the base is going to be the adjacent side and the height of the pole is going to be the opposite side.

What is the trigonometric ratio that connects the adjacent side and the opposite side?

Tan theta is nothing but opposite side upon adjacent side.

In this case we measured our theta to be 60 degrees.

So you will get tan of 60 is the height of the pole let's say is H, H by this distance we measured it as 30 meters so it's H by 30, tan 60 is H by 30 so H will be 30 into tan 60, tan 60 you know is root 3, so the height of the pole is 30 root 3 meters.

Now if you replace this pole with a mountain, doesn't matter, correct?

Doesn't matter, the height of the mountain would be the exact same logic, the exact same way in which you would measure it.

So that's how you use trigonometry to measure very large heights and distances.

You measure 1 distance, you measure 1 distance which is manageable and then measure an angle which helps you measure large heights such as the height of this pole or the height of a peak.


Speaker 0: We have a very, very interesting topic at hand today. 1 that has several real life applications. 1 that you've already learnt. The topic I'm alluding to is Trigonometry. Let's see if we can actually measure the height of a peak.

Or for simplicity, instead of the peak, let's start with a pole. So let's say I have a long pole here and I need to measure its height. Now for the sake of simplicity, I'm going to assume that I'm actually standing inside a hole of sorts and the ground is at my eye level. You'll soon realize why this is important. So the ground is at my eye level, so the base of the pole starts at my eye level and the pole goes on further up from there.

Now we need to measure the height of this pole. How do we do that? Do we see a triangle anywhere here? Actually, we do see 1 side, right? The pole looks like a nice straight line, which is why I picked the pole instead of a mountain.

We need 2 more lines, right? Or 2 more sides to form the triangle. If you join the pole to my eye, If you join the pole to my eye, that could be the second side of the triangle. This is nothing but the distance on the ground, how far away I am from the pole. Now imagine me looking up at the pole and imagine the ray of light that is touching the top end of the pole and reaching my eye.

If you were to imagine that ray of light and you draw a line for it, call it my line of sight, that could be the third side of the triangle, that could be the hypotenuse of the third side of the triangle. 1 side, the vertical side of the triangle is our pole, The horizontal side is the distance between the base of the pole and my eye, and the third side is my line of sight. Perfect. So we have a triangle here. All we need to do is measure 1 side of this triangle.

How do we go about measuring that side? The horizontal distance, if you see, can very easily be measured, right? I could just start at the base of the pole and walk up to here and measure it. It's a distance on the ground. Very simple to measure.

So this distance we can measure. Let's say we measure it and it's 30 meters. Now the height of the pole, say the pole is high, we cannot reach the top, you're not gonna do pole climbing, right? So we cannot practically measure the height of the pole. If we cannot measure the height of the pole, neither can we measure the distance of the hypotenuse because it's the same logic, correct?

Again, you need to reach the top of the pole to be able to measure the hypotenuse. So it's not going to be practically possible to measure the hypotenuse. If we could, then you would simply use Pythagoras' theorem. If you could measure the hypotenuse, you would use Pythagoras' theorem to find out the height. But that doesn't work, right?

If you could measure the hypotenuse, you could very easily measure the height itself. So we cannot measure the hypotenuse. What else can we measure? Now remember we are talking about trigonometry. Trigonometry is something to do with angles.

1 angle is anyway a right angle triangle, which is why we can apply trigonometry in this triangle. Can we measure 1 more angle? Can we measure this angle as I look up at this pole, my line of sight? Can you measure the angle of my line of sight with respect to the horizontal? Can you measure this angle?

Seems difficult, right? Almost impossible. Try looking at objects around. Are you ever able to say what is the angle at which you are looking? Not really, right?

Instead of the naked eye, let me take a binocular. At my eye level. Let me fix it to a post. And the post has a protractor, right? So I can measure the angle at which my binocular is turning.

Now imagine keeping it at 0 degrees. So I'm looking straight down and I'm going to be able to see the base of the pole. I slowly move the binocular up as I see more and more of the pole and I keep moving it up until I finally see the top of the pole. Stop it there. I stop it there.

Now if I measure the angle at which the binocular is with respect to the ground, I can easily measure the angle, I already had a protractor fixed here, right? So I can easily measure the angle, so this angle we now know, right? Let's say it's 60 degrees, so now we know this angle is 60 degrees and the base, this distance, this length is 30 meters. Now in this triangle, now in this triangle can we very easily measure the height? We can very easily measure the height.

It's a right angle triangle, correct? It's a right angle triangle. So 1 side is going to be With respect to this angle, if this angle was angle theta, with respect to this angle, the base is going to be the adjacent side and the height of the pole is going to be the opposite side. What is the trigonometric ratio that connects the adjacent side and the opposite side? Tan theta or cot theta, right?

So let's take tan theta. What is tan theta? Tan theta is nothing but opposite side upon adjacent side. In this case we measured our theta to be 60 degrees. So you will get tan of 60 is the height of the pole let's say is H, H by this distance we measured it as 30 meters so it's H by 30, tan 60 is H by 30 so H will be 30 into tan 60, tan 60 you know is root 3, so the height of the pole is 30 root 3 meters.

We have our answer, right? Now if you replace this pole with a mountain, doesn't matter, correct? Doesn't matter, the height of the mountain would be the exact same logic, the exact same way in which you would measure it. So that's how you use trigonometry to measure very large heights and distances. You measure 1 distance, you measure 1 distance which is manageable and then measure an angle which helps you measure large heights such as the height of this pole or the height of a peak.

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Trigonometry - Measuring Heights And Distances | Class 10 | Learn With BYJU'S