School Quads: Assembly Hall, Science Class, Trig Class and Cafeteria
Get ice cream and stroll to the school
It was another nice summer day with moderately high temperatures made comfortable by occasional breeze – the type of day when most teens like to go out to ice-cream parlors and hang out. Johnny phoned Sara and asked her if they could go for some ice cream. They went together and got the ice cream. Of course Johnny took with him his new bike that he was so proud of. After this, they decided not to hang around there but to go for another walk around their school. One might guess that Johnny had liked their last visit to the school. From that experience, Johnny was sure that his girl friend was not going to waste this chance of pushing him further towards being a Trig whiz. What more could he ask for – being with his sexy lover and his beautiful new bike. Sara loved him and always wanted to be with him. They went for the same stroll on the circular road around their school as the time before.
Sara: Johnny. Do you know that you can get tables of values for Trig functions for different angles? You can also get these values using most calculators. So I brought my calculator with me. We’ll check how accurate your bike is in the North-South and East-West distances.
Johnny: Don’t doubt my beautiful bike. It’s sexy looking. Don’t you think?
Sara said in a playful manner: Sexier than me? You dare not say yes, just shut up. Let’s start at the same spot as the last time. Remember, we started on the circular road around the school at the North-East door of the school. Any time when you tell me our rotation angle in degrees from your bike, I will tell you the expected distances and you can match them on your bike.
They started walking and then Johnny stopped.
My bike says 30° rotation from the original start
Johnny: My bike says 30° rotation from the original start.
Sara: My calculator says that sin 30° is 0.5 and cos 30° is 0.866. Because the hypotenuse is 100 meters, we went 50 (100 times 0.5) meters North and 13.4 meters West (100 – 100 cos 30° or 100 – 86.6 meters).
Johnny: Yup. That’s what my bike says. So the sin and cos thing worked.
Johnny kept going but then he stopped and said 90°. He asked the values for sin and cos.
Values are obvious for sin 90° or cos 90°
Sara laughed out loud. When Johnny asked her what was so hilarious about 90°, she said: I don’t need my calculator for that. We have gone as far North as we could where the height is the same as the hypotenuse of the triangle and that means their ratio is 1 which is sin 90°. We have also traveled 100 meters West. We are exactly on the North of the center of the school which means cos 90° is 0.
Johnny had the picture of the school and this road in his mind and remembered where the map would show him on the Northern most part of the road. He understood the situation to some extent, and kept going. Funny things started to happen. His distance towards North started to decrease but the Westward distance still increased. He showed it to Sara who explained to him with this picture (Fig. 2.1)
What happens around 90° ?
Sara: Remember in Algebra, origin is where the X-axis and the Y-axis cross each other. The East-West line will then be the horizontal axis or the X-axis and the North-South line is the vertical axis or the Y-axis. The height reaches maximum at top of the Y-axis which is an angle of 90°, and then it starts to decrease. The decrease means you are no longer going North but have started to move South. However, you continue moving westward.
Johnny: Now, the bike gives a total rotation of 150°.
Sara: Remember, 150° angle will give you the same height as that of a 30° but in the second quadrant. Here, the values of height are positive but those of the base are negative because you have gone to the left side of the origin.
Johnny: This is confusing. Does that mean sin 150° would be the same as sin 30°?
Sara: Yes, but cos 150° will not be the same as cos 30°.
Johnny: I get it, it will be the same but negative. Cos 30° = 0.866. So, cos 150° = – cos 30° = – 0.866. See, I am no dummy. Does the value of tan also change signs?
After school time cinema
Sara: Yes genius. Now, tell me, if you divide the school into four quadrants, where is the Assembly Hall and who goes there?
Johnny: It would be in the first quadrant. All of us go there for the assembly – the students and the teachers.
Sara: Your Science class is in the second quadrant and the Trig class is in the third.
Johnny: Cafeteria is in the fourth quadrant. That’s my favourite place. Remember, that’s where we met two years ago.
Sara: All go to the Assembly (first quadrant) positively – sin, cos and tan. Only sin is positive in the Science (second) quadrant, only tan is positive in the Trig (third) quadrant and cos in the Cafeteria (fourth) quadrant.
Johnny: That’s a nice way to remember.
Sara: I guess my dad’s school was not round like ours. He said that their teacher told them to remember, “After school time cinema” for all, sin, tan and cos. If you don’t like going to the cinema, you could also say, “After school time chocolates” or “All Silver Tea Cups”. There’s one more thing. Most professionals don’t talk about the angles in degrees. They use radians. The idea is that going around a full circle is 2π radians instead of 360°. Remember, we learned in geometry that π is the ratio of circumference of a circle to its diameter. It has an approximate value of 22/7. So one radian is 2π/360 or about 57°. In that system, the first quadrant is from 0 to π/2, the second from π/2 to π, the third from π to 3π/2 and the fourth from 3π /2 to 2π.
Graphs of trig functions
I guess this was a complicated lesson. Still Sara showed him the graphs for the values of sin x, cos x and tan x for different values of x (Fig. 2.2).
Johnny 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 the 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 sin2x + cos2x =1 or sin2x = 1 – cos2x.
The most intriguing was the graph for tan x. It did not stop at 1 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.
Johnny found it easy to understand the Trig teacher and to do the homework. Johnny and Sara had done all the homework assignments correctly. They both got 100% in the Trig midterm exam too. Johnny was jumping with joy and took Sara out for dinner.
Challenge: Johnny and Sara were just talking and watching TV when they saw a picture of the High Roller Ferris Wheel in Las Vegas. The wheel has a diameter of 520 feet (158.5 meters) and 28 seating boxes spaced apart equally. It rotates around once in 30 minutes. Marvelling at the site, they imagined where they would sit if they were to go there together. Because it is expensive, they would go only for one rotation. They decided that Sara would sit first and then after the wheel had rotated a little bit, Johnny would sit 7 boxes away. When they reached the same height, they would say hi to each other. It would be fun. At what height would that happen? Sara knows it, see if you do.
Solution: There are 28 boxes in the full rotation (360°) of the wheel (Fig.10.3). Sara (S) has already travelled to 7 boxes away (360° x 7/28 = 90°) when Johnny (J) sits in his box. Radius of the wheel is half of the diameter or 158.5 /2 =79.25 meters. Draw a vertical line from C the center of the wheel. S and J will be at equal heights only when the angle SCV = angle VCJ but we know that angle SCV + angle VCJ = 90° and hence the angle VCJ = 45°. Drawing a perpendicular from CJ onto the line CV gives the right angle triangle CVJ with the angle VCJ being 45°. Therefore CV = 79.25 cos 45° = 79.25 x 1/√2 = 56.04 meters. So the total height will be height to the center plus 56.04 meters = 79.25 + 56.04 meters = 135.3 meters.
Note that they could also reach the same height again if they had paid for more than one rotation. Figure this out for yourself.