Another Trigonometric Model Among Temperatures

In 2015, students have followed some daily average temperature registrations in Vicenza, Veneto, Italy. The analysis of another data gathered in previous years convinced the scholars that the daily average temperature follows a trend schematically described by the following function family:

$\large f(t)=\displaystyle{\sin^2\left ( \dfrac{t-6}{4} \right )-\sin \left ( \dfrac{t-6}{4} \right ) \cos \left ( \dfrac{t-6}{4} \right )+a \cdot \cos^2 \left ( \dfrac{t-6}{4} \right )+b}$

where $t$ is time expressed in months of 30 days. For example, on 13 March 2015, 72th day of the year, time $t$ is $\dfrac{72}{30}=2.4$ .

Find the value of parameters $a$ and $b$ knowing that on 13 March 2015 the average temperature was $11.01^\circ C$ and on 26 June of the same year was $32^\circ C$ .

Using the parameters found previously, determine the days when the average temperature was higher than $20^\circ C$ and lower than $30^\circ C$ .

(The temperature is measured in Celsius.)