   Mathematics AI HL
Sample Internal Assessment

Table of content
Rationale
Aim
Background information
Data collection and processing
Process of calculation
Conclusion
Reflection
Bibliography
Appendix

# 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 number. 7/7  0 Words
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## Rationale

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.

## Aim

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).

## Background information

### Climate of mumbai

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.

### Sinusoidal function

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: Figure 1 - Graphical Representation Of A Sine Curve

Similarly, the function could be represented as:

y = f2 (x) = cos cos x

The function could be graphically represented as: Figure 2 - Graphical Representation Of A Cosine Curve

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.

### Complex number

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:

$$r=\sqrt{α^2+β^2}$$

$$\theta=\bigg(\frac{β}{α}\bigg)$$

## Data collection and processing

### Temperature of mumbai

A graphical representation of variation of temperature in Mumbai from 2011 to 2020 is shown below: Figure 3 - Variation Of Temperature In Mumbai From 2011 To 2020

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: Figure 4 - Table On Variation Of Monthly Temperature From 2011 To 2020

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 Figure 5 - Variation Of Average Monthly Temperature From 2011 To 2020

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

### Wind speed in mumbai

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: Figure 6 - Table On Variation Of Wind Speed (Monthly Average) From 2011 To 2020

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 Figure 7 - Variation Of Average Monthly Pressure From 2011 To 2020

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 (0.78+ 2.49) + 13.13

## Process of calculation

### Determination of rate of change of temperature

From section 4.1, the change in temperature in Mumbai over a period of 10 years has been studied. The equation of trend obtained us shown below:

y = 1.42 sin sin (- 2.91) + 29.65

Here, if the temperature is indicated by t and the time (in months) is indicated by m, then the reframed equation would be written as:

t = 1.42 sin sin (m - 2.91) + 29.65…(equation - 1)

The rate of change of temperature with respect to time (in months) could be obtained by differentiating equation (1) with respect to m:

$$\frac{dt}{dm}=\frac{d}{dm}$$ [1.42 sin sin (m - 2.91) + 29.65]

$$\frac{dt}{dm}=\frac{d}{dm}$$ {1.42 sin sin (m - 2.91)} + $$\frac{d}{dm}$$ (29.65)

$$\frac{dt}{dm}$$ = 1.42 × cos cos (m - 2.91) + 0

$$\frac{dt}{dm}$$ = 1.42 × cos cos (m - 2.91) …(equation - 2)

### Determination of rate of change of wind speed

From section 4.2, the change in wind speed in Mumbai over a period of 10 years has been studied. The equation of trend obtained us shown below:

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

Here, if the wind speed is indicated by w and the time (in months) is indicated by m, then the reframed equation would be written as:

w = 4.40 sin sin (0.78m + 2.49) + 13.13…(equation - 3)

The rate of change of wind speed with respect to time (in months) could be obtained by differentiating equation (3) with respect to m:

$$\frac{dw}{dm}=\frac{d}{dm}$$ [4.40 sin sin (0.78m + 2.49) + 13.13]

$$\frac{dw}{dm}=\frac{d}{dm}$$ {4.40 sin sin (0.78m + 2.49)} + $$\frac{d}{dm}$$ (13.13)

$$\frac{dt}{dm}$$ = 4.40 × 0. 78 × cos cos (0.78m + 2.49) + 0

$$\frac{dw}{dm}$$ = 3.42 × cos cos (0.78m + 2.491) …(equation - 4)

### Temperature variation with wind speed (from min to max pressure)

The variation of temperature in terms of wind speed could be obtained by dividing (2) by (4). It would allow to determine the expression of infinitesimally change in temperature due to infinitesimally small change in wind speed. Furthermore, by integrating the expression with respect to wind speed, a generalized equation of temperature compared to wind speed could be obtained.

By (2) ÷ (4):

$$\frac{\frac{dt}{dm}}{\frac{dw}{dm}}=\frac{1.42\times coscos(m-2.91)}{3.42\times coscos(0.78m+2.491)}$$

$$\frac{dt}{dm}\times\frac{dm}{dw}=0.41\times\frac{coscos(m-2.91)}{coscos(0.78m+2.491)}$$

$$\frac{dt}{dm}=0.41\times\frac{coscos(m-2.91)}{coscos(0.78m+2.491)}...$$ (equation - 7)

From (3):

w = 4.40 sin sin (0.78m + 2.49) + 13.13

$$\frac{w-13.13}{4.40}$$ = sin sin (0.78m + 2.49)

arcsin arcsin $$\frac{w-13.13}{4.40}$$ = 0.78m  +2.49

arcsin arcsin $$\frac{w-13.13}{4.40}$$ -2.49 = 0.78m

$$m=\frac{arcsin\ arcsin\frac{w-13.13}{4.40}-2.49}{0.78}$$

m = 1.28 arcsin arcsin $$\frac{w-13.13}{4.40}$$ -3.19…(equation - 8)

From (7) and (8):

$$\frac{dt}{dw}=0.41\times\frac{coscos\bigg(\bigg\{1.28arcsin\arcsin\frac{w-13.13}{4.40}-3.19\bigg\}-2.91\bigg)}{coscos(0.78(1.28arcsin\ arcsin\frac{w-13.13}{4.40}-3.19)+2.491}$$

$$dt=0.41\times\frac{coscos\bigg(1.28arcsin\arcsin\frac{w-13.13}{4.40}-6.1\bigg)}{coscos\big(arcsin\ arcsin\frac{w-13.13}{4.40}\big)}$$

$$t=\displaystyle\int0.41\times\frac{coscos\bigg(1.28arcsin\arcsin\frac{w-13.13}{4.40}-6.1\bigg)}{coscos\big(arcsin\ arcsin\frac{w-13.13}{4.40}\big)}dw$$

### Determination of time period of temperature (yearly)

As an accessory analytical topic, the period of temperature of each year has been calculated and the variation of period of temperature is evaluated in this section. In order to do so, temperature of each month in Mumbai from 2011 to 2020 has been collected (Refer to Table – 1 in Section 4.1, except Mean and Standard Deviation). The variation of temperature has been obtained, graphically, using Desmos Calculator as shown in Section 4.1. However, in Section 4.1, average monthly temperature has been obtained. But in this section, variation of temperature of each year has been calculated using Desmos and hence, the period of temperature variation from 2011 to 2020 would be calculated. Figure 8 - Table On Determination Of Period Of Temperature From 2011 To 2020 Figure 9 - Variation Of Monthly Temperature Of 2011

From Figure 6, the equation of trend could be obtained as:

y = 2.22 sin sin (0.84x - 1.69) + 29.67

Time Period = $$\frac{2\pi}{Angular\ Frequency}$$

$$=\frac{2\pi}{0.84}$$= 7.48 months

Mean = $$\frac{7.48+7.06+…+6.16}{12}$$ = 6.73

$$SD=\sqrt{\frac{(7.48-6.73)^2+(7.06-6.73)^2+…+(6.16-6.73)^2}{12}}$$ = 1.04

Calculation of remaining year are shown in Appendix 1. Figure 10 - Variation Of Time Period Of Temperature From 2011 To 2020

Analysis

From the above graph, variation of time period of temperature every year from 2011 to 2020 could be studied. It is clear that the temperature, though trigonometric in nature, is not having a definite time period every year. It is obtained from the graph that temperature being affected by a lot of environmental as well as other factors, cannot follow a definite trend. However, the values of time period of temperature obtained for each year from 2011 to 2020 is very close to each other with a minimum value of time period obtained as 5.61 months (2014) and a maximum value of time period obtained as 9.1 months (2016). Hence, the overall time period of temperature obtained from 2011 to 2020 is 6.73 months. This indicates that a definite pattern of temperature repeats every 6.73 months in Mumbai. It should be noted that Mumbai does not experience a significant winter. Hence, the time period of temperature is less than 12 months. However, this time period is not same for all cities as the temperature varies differently in different cities.

## Conclusion

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).

The temperature and wind speed varies as trigonometric sinusoidal function and the relationship between temperature and wind speed is an integral with a ratio of two trigonometric functions.

• Temperature varies with wind speed as an integral with a ratio of two sinusoidal function.
• The overall equation of temperature obtained by considering monthly average temperature from 2011 to 2020 is: t = 1.42 sin sin (m - 2.91) + 29.65 where, t indicates temperature and m indicates time (in months).
• The overall equation of wind speed obtained by considering monthly average wind speed from 2011 to 2020 is: w = 4.40 sin sin (0.78m + 2.49) + 13.13 where, w indicates wind speed and m indicates time (in months).
• The equation of temperature compared to wind speed: $$t=\int0.41× \frac{coscos\big(1.28arcsin\ arcsin\frac{w-13.13}{4.40}-6.1\big)}{coscos\big(arcsin\ arcsin\frac{w-13.13}{4.40}\big)}$$ dw where, t indicates temperature (in ) and w indicates wind speed (in kmph).
• Overall period of temperature obtained is 6.73 months, i.e., same temperature repeats at an interval of 6.73 months.
• Temperature is affected by a lot of environmental factors and other man-made factors such as carbon emission, which results in obtaining different time period as the pressure, humidity, carbon emission is different in different years.

## Reflection

### Strength

• The values of temperature, and wind speed are obtained from data sources which are authenticated. It has made the increased the strength of the obtained relationship and reliability of the exploration methodology.
• As the range of data collection is very extensive, i.e., long period of 10 years from 2011 to 2020, it has enabled the exploration and the obtained trend to match up with the real variation of temperature, making the exploration more coherent.
• The tools used in calculation are Desmos and TI – Nspire CAS CX, both of which are verified by IB.
• Standard Deviation has been calculated for each data table and values obtained are very less. This makes the exploration error free and more coherent.

### Weakness

• Verification of data obtained from the secondary data sources are not possible which could incur some discrepancy in the exploration.
• The indefinite integral obtained in the relationship between temperature and wind speed was not solved as the integral could not be solved using calculator and solution of such integral is also beyond the scope of IB.

### Future scope

To further explore on this topic, the effect of humidity on temperature could be studied. It is obtained in this exploration that temperature is affected by a lot of environmental factors. Thus, the relationship between temperature and humidity, which is a significant environmental factor would be an important exploration. This exploration will involve two phases. In first phase, data comprise temperature and humidity of a particular city should be collected over a long period of time. It would also include derivation of equation of trend of temperature and humidity over months. In the second phase, the relationship between temperature and humidity would be determined using the previously obtained equation of trend. The aim of the exploration could be framed as: To determine the relationship between temperature and humidity of Mumbai based on the observation of temperature and humidity of last 10 years (2011 to 2020).

## Appendix

• Calculation of period of temperature from 2012 to 2020: Figure 11 - Variation Of Monthly Temperature Of 2012

From Figure 11, the equation of trend could be obtained as:

y = 2.01 sin sin (0.89x - 2.14) + 29.50

Time Period = $$\frac{2\pi}{Angular\ Frequency}$$

$$=\frac{2\pi}{0.89}$$ = 7.06 months Figure 12 - Variation Of Monthly Temperature Of 2013

From Figure 12, the equation of trend could be obtained as:

y = 2.10 sin sin (0.85x - 1.71) + 29.33

Time Period = $$\frac{2π}{Angular\ Frequency}$$

$$=\frac{2\pi}{0.85}$$ = 7.39 months Figure 13 - Variation Of Monthly Temperature Of 2014 Rom Figure 13, The Equation Of Trend Could Be Obtained As

From Figure 13, the equation of trend could be obtained as:

y = 2.43 sin sin (1.12x + 2.82) + 3 0.09

Time Period = $$\frac{2\pi}{Angular\ Frequency}$$

$$=\frac{2\pi}{1.12}$$ = 5.61 months Figure 14 - Variation Of Monthly Temperature Of 2015

From Figure 14, the equation of trend could be obtained as:

y = 2.02 sin sin (1.05x - 3.08) + 30.00

Time Period = $$\frac{2\pi}{Angular\ Frequency}$$

$$=\frac{2\pi}{1.05}$$ = 5.98 months Figure 15 - Variation Of Monthly Temperature Of 2016

From Figure 15, the equation of trend could be obtained as:

y = 2.20 sin sin (0.69x - 1.19) + 29.59

Time Period = $$\frac{2\pi}{Angular\ Frequency}$$

$$=\frac{2\pi}{0.69}$$ = 9.10 months Figure 16 - Variation Of Monthly Temperature Of 2017

From Figure 16, the equation of trend could be obtained as:

y = 2.14 sin sin (1.02x - 2.61) + 30.03

Time Period = $$\frac{2\pi}{Angular\ Frequency}$$

$$=\frac{2\pi}{1.02}$$ = 6.16 months Figure 17 - Variation Of Monthly Temperature Of 2018

From Figure 17, the equation of trend could be obtained as:

y = 1.91 sin sin (0.99x - 2.59) + 30.10

Time Period = $$\frac{2\pi}{Angular\ Frequency}$$

$$=\frac{2\pi}{0.99}$$ = 6.34 months Figure 18 - Variation Of Monthly Temperature Of 2019

From Figure 18, the equation of trend could be obtained as:

y = 1.58 sin sin (1.04x - 3.13) + 29.24

Time Period = $$\frac{2\pi}{Angular\ Frequency}$$

$$=\frac{2\pi}{1.04}$$ = 6.04 months Figure 19 - Variation Of Monthly Temperature Of 2020

From Figure 19, the equation of trend could be obtained as:

y = 1.58 sin sin (1.02x - 3.09) + 28.99

Time Period = $$\frac{2\pi}{Angular\ Frequency}$$

$$\frac{2\pi}{1.02}$$ = 6.16 months