Investigating the trigonometric functions to describe the variation of Temperature against Time over a period of 10 years, the variation of Wind Speed against Time over a period of 10 years and finding a relation between the two using the concepts of differential calculus and complex numbers.

Education has the power to light the flame of development and build resources from scratch. Being an inquirer and a creative thinker, I always aspire to construct ways to simplify the complexities and difficulties of life and society around us. IB has always inspired me to work on real life scenarios by taking case studies that has real life significance so that the curricular learning could be implemented.

My parents’ working habits and lifestyle have also pushed me towards social work, engaging myself wherever there is any local or global issue. As global warming is ringing the bell of devastation, if it is not dealt with care and perfection, now, it would gift a future with pain and trouble to our future generations. This has pushed me to investigate the effect of various parameters that affects the temperature of a city as temperature is the face of global warming.

By researching on the topic and also reading a few on environmental studies such as ‘The Journal of Environmental Studies and Sciences’, ‘International Journal of Environmental Studies’, and many more, I deciphered many parameters that directly or indirectly affects the temperature. By researching more on this topic, pages became repetitive as the most of the resources are on relationships between the direct parameters of global warming and the parameters affecting it. It has allowed me to think on the factors which are still unknown. I thought, “Does wind affect the environmental temperature?” “If it affects temperature, then how does it affect the temperature?” It has pushed me to interpret the temperature in reference to wind current.

When it comes to comparison between two entities or determining the effect of any parameter on other in a quantitative aspect, no subject is as helpful as Mathematics. As variation of temperature is consistent all through the year, I did a revision of Topic 3 of IB Mathematics – Geometry and Trigonometry because only trigonometric functions may represent the variation of temperature. Finally, with collecting a few more secondary data resources, I landed on the aim of the exploration.

The prime motive of this investigation is to determine the relationship between temperature and wind speed of Mumbai based on the observation of temperature and wind speed of last 10 years (2011 to 2020).

Mumbai is located at the western coast of India and one of the most urban cities in the India. Due to its location, the city experiences a chain of land and sea breeze all through the year. It offers a significant wind chain apart from the wind belts that blow through Mumbai.

Furthermore, India is situated near the equator and lies within the scope of a subtropical country. Hence, the population of Mumbai experience significantly high temperature all through the year.

Mumbai is also one of the busiest cities in India. Due to the emission of carbon in the form of car fumes, industrial fumes and many more, Mumbai experiences significant air pollution. As a result, temperature is increased. Mumbai experiences a temperature between 25℃ to 40℃ in summer and between 17℃ to 30℃ in winter.

A trigonometric function which varies in terms of Sine or Cosine of the independent variable is known as a Sinusoidal Function. From the definition, if the function varies with respect to x:

y = f₁ (x) = sin sin x

The function could be graphically represented as:

Similarly, the function could be represented as:

y = f₂ (x) = cos cos x

The function could be graphically represented as:

A generalized form of a sinusoidal wave is represented by:

y = f(x) = p×sin sin (qx + r)  + S

where,

p = Amplitude

q = angular frequency

r = phase

s = Vertical shift

Amplitude is the magnitude of y with respect to the equilibrium position. Angular frequency may be defined as a factor the number of patterns or cycles that takes place in 1 second is multiplied. Phase may be defined as the value by which the function has been translated horizontally (along X – Axis). Vertical shift is the distance by which the equilibrium of central axis of the waveform has been shifted from the origin.

A complex number may be defined as the number which has a real part (real number) along with a number which involves calculation of square root of negative 1. The distance between any point (complex number) represented on Argand Plane with respect to the origin is known as the modulus (r) of a complex number. Argument (θ) of a complex number may be defined as the angle of the slope of the point (complex number) on the Argand Plane. If a complex number is represented as: z = α + iβ, where z is the complex number and α,β are real constant values. Then:

\(r=\sqrt{α^2+β^2}\)

\(\theta=\bigg(\frac{β}{α}\bigg)\)

A graphical representation of variation of temperature in Mumbai from 2011 to 2020 is shown below:

From Figure 3, it could be said that the temperature is a periodic function and though the function is not perfectly periodic as periodic function is a function which repeats its values at definite interval of time. Here, the variation of temperature repeats the same nature at a definite interval of time, however, same values of temperature are not repeated. Variation of temperature is caused by a lot of parameters which are not within the scope of this exploration. Hence, to have an overall idea of variation of temperature, the average temperature of each month over a period of 10 years from 2011 to 2020 is obtained:

Sample Calculation:

 \(Mean \, \,Temperature \, \,of \, \,January =\frac{28+29+…+27}{12}= 28.2℃\)

\(Standard \, \,Deviation \, \,of \, \,January = \sqrt{\frac{(28-28.2)^2+(29-28.2)^2+…+(27-28.2)^2}{12}} = 0.63\)

Using Desmos Graphical Calculator, the equation of the best fit curve obtained from the values as shown in Table 1:

y = 1.42sin sin (x - 2.91) + 29.65

It is known that the wind speed is a periodic function and though the function is not perfectly periodic as periodic function is a function which repeats its values at definite interval of time. Here, the variation of wind current repeats the same nature at a definite interval of time, however, same values of wind speed are not repeated. Variation of wind current is caused by a lot of parameters which are not within the scope of this exploration. Hence, to have an overall idea of variation of wind speed, the average wind speed of each month over a period of 10 years from 2011 to 2020 is obtained:

Sample Calculation:

\(Mean \, \,Pressure \, \,of \, \,January =\frac{9.5+11.4+…+13.6}{12}= 10.85 \,km. hr^{-1}\)

\(Standard \, \,Deviation \, \,of \, \,January = \sqrt{\frac{(9.5-10.85)^2+(11.4-10.85)^2+…+(13.6-10.85 )^2}{12}} = 1.63 \, \,km. hr^{-1}\)

Using Desmos Graphical Calculator, the equation of the best fit curve obtained from the values as shown in Table 2, is shown below:

y = 4.40 sin sin (0.78x + 2.49) + 13.13