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

## About the first trip to Connaught Place

Tanya, a 16 year old smart girl from Patna has come to spend the Dussehra holidays with her maternal aunt in Delhi. The aunt has a son Adi who is about the same age as Tanya. Tanya treats him as her brother and sends rakhi to him every year. At Tanya’s request, he took her to visit Connaught Place on the condition that she would help him with learning Trigonometry. Anyway, both went to Connaught Place and now the whole family is sitting together for an evening tea.

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

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

Tanya: Connaught Place 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 took Metro back to home.

Adi: She also bought me a shirt. She taught me Trig when we there and more after we came home. Dad, it all went over my head when they taught Trig in school and in the tution center. When Tanya taught it, I found the subject to be very easy and it all made sense. If Tanya teaches me a little more, I can quit the tution center for ever.

Adi’s dad: Really !

Tanya: Uncle, I think that Connaught Place was built for teaching Trig – its three concentric circles and thoughtfully made radial roads attest to it. You know, we are going tomorrow again.

## Connaught Place Again

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 for a while and then Tanya turned the App on her smart phone. This way, they would know how much they move in any given direction. She told Adi that today he should keep watching the distances moved using this smart phone. Like yesterday, they walked east until they reached Connaught Circus and Odeon sweet house (see picture).

Adi: We have moved 350 meters towards East and zero towards North.

Both kept walking anticlockwise on Connaught Circus while chatting and looking around. Slowly, they reached Radial Road 4 which is North of Connaught Circus. On the way to there, Adi had checked on the cell phone that the distance they moved towards East had been gradually decreasing while for movement towards North it had been increasing.

Adi: Now, we have reached Radial Road 4. Our North movement is 350 meters and our movement towards East has become zero. Let’s see what happens next. Crap, looks like something is wrong with the cellphone. Now the distance moved towards North is decreasing, and even worse, the distance moved towards East is negative.

Tanya: There is nothing wrong with the cellphone, and I expected this to happen. On Connaught Circus, we went as far as North as we could go in this circle. Now we are going towards South, so the distance moved towards North decreases. Adi, do you remember when we made X and Y axes on a graph paper. The vertical axis was Y and the horizontal axis was X. We have gone as high on the Y-axis as we could and will come down now. Also, do you remember the values of the X-axis points to the right of the Y-axis were positive and those towards its left were negative (see this picture). See I have made four quadrants (quads) in this picture. First quad is 0° to 90°, second is 90° to 180°, third is 180° to 270°, and the fourth is 270° to 360°. We can also say that the values of X and Y are positive in the first quadrant but in the second quadrant those of X are negative. So after a rotation of 90° in a circle, in the second quadrant we start going towards West which is negative of going East.

Adi kept walking along Connaught Circus. Distance moved towards North from the starting point kept decreasing and the negative value of the Eastward movement kept increasing. Then they crossed Shahid Bhagat Singh Marg which is the Westernmost point of Connaught Circus. That meant that they had entered the third quad. He kept going and reached the fourth quad when they crossed Radial Road 1. By this time he was confused and was not paying much attention but the App was still on and recording. They kept moving to the starting point – Odeon sweet house.

Adi: Tanya, I am understanding the movements a little bit but what do they have to do with Trig ?

Tanya: You will understand everything at home. For now, I want some ice-cream.

They found an ice cream parlor, ate some ice cream and them went home.

At home, Tanya made a sketch and then called Adi.

Tanya: Do you remember that in algebra, we made horizontal and vertical axes. Values on the horizontal axis were positive on right hand side of the vertical axis and negative on the left. Same way on the vertical axis the values were positive above the horizontal axis and negative below.

## The four quadrants and X-Y axes

Adi: Tanya, you even made four quads and a right angle triangle in each quad. The hypotenuse in each triangle is 350 meter. That way∠BAC = 30°, ∠BAE = 150°. ∠BAF = 210° and ∠BAG = 330°.

Tanya: Now determine the values of the Trig functions of these angles.

Adi: sin BAC = 0.5, sin BAE = 0.5, sin BAF = – 0.5 and sin BAG = -0.5. cos BAC = 0.866, cos BAE = -0.866, cos BAF = -0.866 and cos BAG = 0.866. tan BAC = 0.577, tan BAE = – 0.577, tan BAF = 0.577 and tan BAG = – 0.577. I remembered the values for those in the first quad from yesterday, the rest I just changed the signs based on what you told me. They were all positive in the first quad, only sin was positive in the second, tan in the third and cos in the fourth. Now I understand what you were trying to see in Connaught Place. It is interesting but I don’t know how I will memorize them.

Tanya: My dad taught me “after school time cinema” or “after school time chocolates or “All spoon tea cups”. Memorize what ever you like. I like chocolates. One more thing.

In professional work, angles are usually measured in radians rather than degrees. The principle is that it takes 2π radians or 360° in completing the turn of a circle. Remember we learnt that the ratio of the circumference of a circle to its diameter is π. π is approximately 22/7. Therefore a radian is approximately 360°/ 2π ≈ 57°. In this system the first quad is 0 to π/2, second from π/2 to π, third from π to 3π/2 and the fourth quad is 3π/2 to 2π.

May be this lesson was difficult for Adi. Still Tanya gave him graphs for the values of sin x, cos x and from 0 to tan x from 0 to 2π.

Adi noticed that the values of sin x and cos x in the graphs ranged only from 0 to 1. That made sense because the height or base of a right angle triangle cannot exceed the length of its hypotenuse. Also, it was interesting that sin x was positive only in the first two quadrants and cos x in the first and fourth. He also noticed that cos x decreased with the increasing sin x. That also made sense because sin^{2}x + cos^{2}x =1 or sin^{2}x = 1 – cos^{2}x.

The most intriguing was the graph for tan x. It did not stop at one like the sin x and and the cos x graphs did but kept increasing in its value. Also, it was positive just before 90°, and then suddenly it took a very large negative value. The same thing happened at around 270°. It reminded him of a guy who would be extremely happy one moment and then suddenly start crying.

Adi kept himself busy with the Trig book. This assurance was enough that Tanya would help with something if needed. After two days Tanya returned to Patna.

After a month, Adi sent her a picture on the phone. It was the result of his Trig test in which he had scored 100 %. We don’t know who all Tanya showed this picture to: to say how smart her brother was or to say how smart she had made him. Well you decide.

## Challenge

Tanya sees Trig in everything. See what she did now, she gave Adi a picture of Qutab Minar and told him that it has a height of 93 meters. She drew a straight line AC downward along the wall, and then drew a line AB vertical to the ground. She drew a line connecting B and C, and said it was 5.8 meters long. She then asked him two questions. One was the length of the wall from top to bottom and the second was the measurement of the angle BCA. Can you figure these out ?

Answer

Length of the wall based on Pythagoras theorem will be √(93^{2}+5.8^{2}) = 93.18 meters.

Now tan BCA = 93/5.8. Therefore the angle BCA = arctan (93/5.8) = arctan 16.03 which is 86.43° (From the Internet).