### (There are a number of places with this name but this story is about the Connaught Place in New Delhi, India)

## Connaught Place an economic and business center

Connaught Place is probably the biggest economic and business center in Delhi – the capital of India. Not just that, Indian and foreigner tourists find this to be one of the biggest attractions of this city. For visitors from Europe and America, it is a comfortable market. Mind you, college and high school students also enjoy roaming here. This area was built at the time when the British rulers moved the Indian capital from Calcutta to New Delhi. Around this area, particularly to the South there are the mansions which were built for the British officers and then kings of Indian colonies. Connaught Place was built as a graceful market place for them. Further to its South there are the legislative and administrative centers of the capital.

## The hallmark of Connaught Place is the appearance of a large circle

In today’s Delhi, Connaught Place is in the center or somewhat to the East of the center of the city. It is surrounded by major population areas in all directions. The hallmark of Connaught Place is the appearance of a large circle from which roads radiate out in different directions. The primary design of this area are three concentric circles. In the middle is Center Park which has places for people to sit and enjoy the scenery. This park has been used for various types of gatherings. This 150 meter radius Park is encircled by a road which is termed Inner Circle. About 100 meters away from this road is Middle Circle and another 100 meter is the outermost road which is termed Connaught Circuit. Radial roads numbered 1 to 7 and Parliament Street come out from Connaught Circuit. Inbetween the different roads are shops, restaurants and cafes, banks, hotels, movie theaters etcetera. An aerial view of this area shows a distinct symmetry. And yes, there are many sites worth seeing on the East and South of this area. Being attracted by the beauty of this area, Bollywood has also used it for filming. There are several modes of transportation to reach this area. No wonder, Connaught Place is crowded all the time.

## Tanya and Adi

Tanya is a 16-year old girl living in Patna, India. She is the sole offspring of her parents and has come to Delhi to visit her mausi (mother’s sister). This is the first time she has made an unaccompanied trip to Delhi. Last time she came to Delhi was 6 years ago when she was just ten years old, and came here with her mom. Mausi has a son Adi who is the same age as Tanya. Tanya sends rakhi to Adi and treats him as her brother (a symbolic ritual that binds a brother and sister). Occasionally the two of them also talk over the phone. When she came to Delhi, Tanya knew that she would have his company. The two of them were sitting and chatting about whatever, your guess is as good as mine.

Tanya: I have not seen Connaught Place yet. A friend of mine from Patna saw it and said it rocks – a must see place. Can you take me tomorrow to see it, please pretty please ?

Adi: It is difficult. I am weak at Math and everyday I go to a tuition college. I have to homework from there too. My mom will not give me permission for loafing around like this.

Tanya: I will get you the permission. I am the favorite niece of your dad. On my persuation, he will ask you to do it. By the way, what subject in Math is the course for which you go to the tuition center ?

Adi: Trigonometry, it all goes over my head.

Tanya: Adi, in that case put a smile on your face. I will make you such a genius in Trig that you will be teaching your friends. You will not even have to go to the tuition center. Only, tomorrow and the day after you will have to come with me and show me Connaught Place. Don’t panic, I will get you the permission and I also have some money – my dad gave me some for the trip.

Sticking to her words, Tanya went to Adi’s dad: Uncle, what is going on here ? Why are you torturing my brother Adi by sending him to the tuition school ? I can easily teach him Trigonometry very well, some now and then some on Skype from Patna. Get him out of this suffering and tell him to spend the next four days with me, his sister. He can take me to see places in Delhi. His mind will be refreshed and you will also save the tuition center fee.

That was all it took. Adi’s dad knew that Tanya got near perfect marks in every subject. He thought that Adi might learn something from being in her company. He went to his son: Adi, your sister has come after 6 years. I don’t know when she will come again. Talk to her. Here is some money. Take her to see places in the city. You can take a few days off from the tuition school.

Adi said yes to his dad, put the money in his pocket, and then talked to Tanya: Hey sis, what kind of magic did you use on my dad. He almost ordered me to take you around to show you Delhi, and even gave me some money for the expenses. Okay, tell me one thing. What made you so confident to say that you could make me a genius in Trig ?

Tanya: First, I got 100% mark in this subject, second I have helped many of friends in it and third my dear brother, just for you I have installed a special App on my smart phone. So what time shall we go tomorrow for a visit to Connaught Place ?

## Connaught Circus is the outermost circular road of the Connaught Place Area

Next morning, Adi’s mom filled their stomachs after which Adi and Tanya left home. They caught Metro and reached Connaught Place. They sauntered around in the Central Park area for a while and then walked over to the eastern part of Connaught Circus which is the outermost circular road. This spot was near Odeon sweet house. There, Tanya took out her smart phone.

Adi: Do you want to call home ?

Tanya: No, I have turned on the App I told you about. It uses GPS to determine and record our location. As we move along Connaught Circus, it will keep recording our location and will also tell us how much we have moved towards North and how much towards West.

They moseyed along while chatting and watching the scenes around them. Slowly while walking, they reached Radial Road 4 which is the North of Connaught Circus. There were all sorts of shops there. Tanya started looking for presents for her friends but Adi stopped her saying that they could get the same items a lot cheaper in Palika Bazar or on JanPath. They walked from there to Janpath. The place had all sorts of vendors. Tanya bought some things from there. Now that they were hungry and thirsty, they found a cafe nearby and had some refreshments. They took Metro and went back to home.

Adi’s mom had lunch ready. They ate while chatting and then decided to rest for a while. What rest ? Tanya transferred all the App generated data onto her laptop, and started playing with it. Actually, she was analyzing the data but for Tanya it was just a game.

Tanya: Adi.

Adi: What ?

Tanya: See this map of Connaught Place.

Adi: Nothing new, I have seen it a million times.

## The schematic map

Tanya: Okay, then see this schematic map. It shows all the roads and the buildings. Remember, I turned on the App at Odeon sweet house. It shows that too. We walked anticlockwise on Connaught Circus and then at Radial Road 6, Bhape da hotel, Radial Road 5 and Radial Road 4, I measured how much North and how much West we had gone from Odeon sweet house. All these spots are also shown on this schematic map.

Adi: It looks good. What else ?

## Trigonometry is about right angle triangles

Tanya: The next figure is only for the area we covered today. In this picture, a straight line AB has been drawn from the Central Park to Odeon sweet house where we started recording our movement. Another line AC has been draw from the center to Radial Road 6. From C, I have a line CB vertical to AB. Because from Odeon sweet house to Radial Road 4 forms one fourth of a circle which is 90°, and the rotation until Road 6 is one third of that, the angle BAC is 30°. According to my smart phone, we had moved 175 meters North (line BC) of Odeon sweet house. Also according to this map AC is 350 meters.

Adi: The ratio of the height of a right angle triangle to its hypotenuse is the function sin, so the ratio of the lengths of CA and AC will be called sin BAC. According to you sin BAC is 175/350 or 0.5. Hey, my calculator shows that sin 30° is 0.5. So how much have we moved westwards according to your App ?

Tanya: 46.9 meters. That means AB = 350 – 46.9 or 303.1 meters. The ratio of the lengths of the base and hypotenuse of a right angle triangle is the function cosine. So cos (BAC) = 303.1/350 or 0.866.

Adi: Yes, that’s what my calculator shows.

Tanya: Actually, we did not need to determine the length of AB this way. Do you remember the Pythagoras theorem from geometry?

Adi: Yes. In a right angle triangle base^{2} + height^{2} = hypotenuse^{2}.

Tanya: Now, what will you get if you divided both sides of this equation by hypotenuse^{2 }?

Adi: height^{2}/hypotenuse^{2 }+ ^{ }base^{2}/hypotenuse^{2 }= 1. My god, this means sin^{2}x + cos^{2} x = 1. No one ever taught it to me like this. Both, the school sir and the tuition teacher, just told me the formula and asked me to memorize it. Maybe that is why it didn’t occur to me that I could use it here.

Tanya: Yes, this also gives you cos x = ±√(1 – sin^{2}x). So we can calculate cos x from sin x. What are all the trigonometric functions you know?

Adi: tan, cosec, sec and cot. Height/base is tan x. cosec x = 1/sin x, sex = 1/cos x and cot x = 1/tan x.

Tanya: If you know the value of any one of the trigonometric functions of an angle, you can determine all the others because they are all related. Now, let’s see sin 30° = 0.5. What will be the values of all the other functions ?

Adi; You made everything so simple. Because sin 30° = 0.5, cos 30° =√(1 – 0.5^{2}) = 0.866, Then tan 30° = sin 30°/ cos 30° = 0.5/0.866 = 0.577. All the others are reciprocals which one can easily calculate using a calculator.

Adi: Sis, you have also drawn the right angle triangles ADE and AFG in this picture, what for ?

Tanya: Yes, I wanted to see how much attention you have been paying. Tell me what will be the values of sin DAE and sin FAG ?

Adi: Regardless if the sizes of the triangles these angles will be 30° and hence sin DAE and sin FAG will all be 0.5. If you tell me the lengths of the hypotenuse of one triangle, I can tell you its height and base.

Tanya: The hypotenuse of the middle one is 250 meters and that of the small one is 150. There is no need of determining their heights and bases. I just wanted to increase your self-confidence.

Adi: Is that it or there is more ?

Tanya: We just started. Look at this second picture. When we reached Bhape da hotel, we had gone 247.5 meters North. In this right angle triangle ABC, the hypotenuse will remain the same as it is the radius of Connaught Circus which is 350 meters. As per my App the base of this triangle is also 247.5 meters. Now, tell me values of all the trigonometric functions of the angle BAC.

Adi: This is an isosceles triangle because AB = BC. Therefore, angle BAC = (180° – 90°)/2 = 45°. Therefore,

sin 45° = cos 45° = 247.5/350 = 0.707, tan 45° = 0.707/0.707 = 1

cosec 45° = sec 45° = 350/247.5 = 1.414, cot 45° = 0.707/0.707 = 1

Tanya: Adi boy, you know Trig very well.

Adi: No, what you are making me do is geometry which I know very well.

Tanya: So, you are good at geometry. Trig is the same as geometry except that instead of writing long statements, you just write the values of the functions. The relationship between the two is similar to that between arithmetic and algebra. Okay, do the last problem of today. When we reached Radial Road 5, we had gone 303.1 meters North of Odeon sweet house. Knowing this, determine values of all the functions of BAC in this picture.

Adi: Since at that time we had gone two thirds of the first quarter of a circle, angle BAC =60°. AC will remain 350 meters. Therefore, sin BAC = 303.1/350 = 0.866. Tanya, can you please check if sin 60° = 0.866 ?

Tanya: Yes, and that means you were correct in saying that angle BAC = 60°.

Adi: Thank you Tanya. Now, because sin BAC = 0.866, cos BAC = √(1 – sin^{2} BAC) = 0.5. The tan BAC = 0.866/0.5 = 1.732. Should I also determine the reciprocals of these three functions?

Tanya: No that’s enough for today. We will go to Connaught Place tomorrow again and then do more. For now, let’s go chat with everyone. Maybe that way we will get some tea.

Adi: Tanya, you have also put a star of Radial Road 4. You cannot make a triangle from there or make one with zero base but the same height as the hypotenuse. So shall we say that sin 90° = 1, cos 90° = 0 and tan 90° = ∞ ?

Tanya: Adi, you turned out to be very smart in finding out all of that on your own.

The whole family sat together for the tea.

Adi’s dad: Tanya, tell us how you liked Connaught Place.

Adi’s mom: Tell us in detail like where you two went, what you ate and what you purchased.

Tanya: Connaught Place is an interesting place, We left Metro and went towards Odeon sweet house, and from there we strolled on Connaught Circus towards North all the way to Radial Road 4. Then we walked to Janpath where I bought some souvenirs for my friends in Patna. After that we went to a cafe and then back to Central Park and then took Metro back to home.

Adi: She also bought me a shirt. She taught me Trig when we were there and more after we came home. Dad, it all goes over my head when they teach Trig in school and in the tuition center. But when Tanya taught it, I found the subject to be very easy, and it all made sense. We will go to Connaught Place tomorrow again.

## Challenge

Ramjas School No. 2 in Delhi is on the top of the Anand Parbat mountain. Many students come to that school from the Karol Bagh side – inluding Munna. Munna comes there riding on an old beat up scooter. He tells his dad that his scooter is like a jalopy. While going on Ghati Road, for 350 meters he has to walk and drag the scooter because the road is too steep, but afterward the slope decreases and he can ride the scooter again. Ghati road is 231 meters above the sea level when it starts but after 350 meters, it is at 318 meters above the sea level. Instead of sympathizing with poor Munna, the math teacher asks him to make all the Trig functions of this climb.

Answer

Construct a right angle triangle ABC whose hypotenuse AC is 350 meters and height BC is 318 – 211 = 87 meters. Draw a horizontal line AB as base of the triangle. Now, compute all the Trig functions of the angle BAC.

cos BAC = √(1 – sin^{2} BAC) = 0.97, sec BAC = 1/0.97 = 1.03

tan BAC = sin BAC/cos BAC = 0.26, cot BAC = 3.89.

Arcsine is the inverse of the function sin. The angle BAC can be determined as arcsin 0.25 = 14.45° either using Trigonometric Tables or a calculator.