The daylength on the first day of each month from January $2021$ to January $2023$, $D$ (in ho

Question

The daylength on the first day of each month from January $2021$ to January $2023$, $D$ (in hours), for four cities of different latitude is recorded and sketched with respect to $t$ (in months) using smooth curves in the same diagram below.

Graph of daylength D in hours vs time t in months for Singapore, Harbin, Reykjavik, and Sydney

Explain with clear reasons why sine and cosine functions can be used to model the above data.
The curves representing the daylength of Singapore (respectively, Harbin and Reykjavik) can be approximated by $D_S = a_1 \cos b_1 t + c_1$ (respectively, $D_H = a_2 \cos b_2 t + c_2$ and $D_R = a_3 \cos b_3 t + c_3$). State, by giving clear reasons, the relationships between $a_1$, $a_2$, $a_3$; $b_1$, $b_2$, $b_3$ and $c_1$, $c_2$, $c_3$, respectively.
Given that the daylength of Sydney on $01$ January $2021$ is $14$ hours (nearest integer), suggest a suitable function to model the daylength of Sydney for $0 \le t \le 24$.
Using your answer in (iii), estimate the daylength of Sydney on $01$ July $2025$.
The table below shows the latitudes at which the four cities locate. State the relationship between the amplitude of the daylength functions and the latitude of the city. Hence make a conjecture on the amplitude of the daylength function at the North Pole.

Singapore (Singapore)	Harbin (China)	Reykjavik (Iceland)	Sydney (Australia)
$1^{\circ} 17'\mathrm{N}$	$44^{\circ} 20'\mathrm{N}$	$64^{\circ} 09'\mathrm{N}$	$33^{\circ} 55'\mathrm{S}$

Data Source: [Sunrise Sunset Times Lookup - MAPLOGS](https://www.maplogs.com/)

Accepted Answer

(ii) $|a_{1}| < |a_{2}| < |a_{3}|, b_{1} = b_{2} = b_{3} = \dfrac{\pi}{6}, c_{1} = c_{2} = c_{3} = 12$; (iii) $D_{syd} = 2\cos \left(\dfrac{\pi t}{6}\right)+12$; (iv) $10$ hours; (v) amplitude increases with distance from equator, North Pole amplitude about $12$ hours

Trigonometry - Trigonometric Graphs and Parameters

Trigonometry - Trigonometric Graphs and Parameters

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