6. (a) Express sin x + 2 cos x in the form R sin (x + a) where R and a are constants, R > 0 and 0 < a < \frac{\pi}{2} Give the exact value of R and give the value of a in radians to 3 decimal places - Edexcel - A-Level Maths Pure - Question 8 - 2020 - Paper 1

Using the equation given for the temperature:

$\theta = 5 + \sqrt{5} \sin\left(\frac{\pi}{12} + 1.107 - 3\right)$

The maximum temperature occurs at the maximum value of the sine function, which is 1. Therefore:

$\theta_{max} = 5 + \sqrt{5}(1) = 5 + \sqrt{5} \approx 7.236 \text{ degrees Celsius}$

To find the time 'i' when the maximum occurs, we set:

$\frac{\pi}{12} + 1.107 - 3 = \frac{\pi}{2}$

This gives:

$\frac{\pi}{12} - 3 = \frac{\pi}{2} - 1.107 \implies i = \frac{\pi}{12} + 1.107 - 3$

After solving,

$i \approx 13.2$

Converting this to standard time gives:

• Either 13:14 or 13 hours 14 minutes after midnight.