In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
Solve, for \(0 < x \leqslant \pi\), the equation
$$5 \sin x \tan x + 13 = \cos x$$
giving your answer in radians to 3 significant figures.
The temperature inside a greenhouse is monitored on one particular day.
The temperature, \(H ^ { \circ } \mathrm { C }\), inside the greenhouse, \(t\) hours after midnight, is modelled by the equation
$$H = 10 + 12 \sin ( k t + 18 ) ^ { \circ } \quad 0 \leqslant t < 24$$
where \(k\) is a constant.
Use the equation of the model to answer parts (a) to (c).
Given that
the temperature inside the greenhouse was \(20 ^ { \circ } \mathrm { C }\) at 6 am
\(0 < k < 20\)
(a) find all possible values for \(k\), giving each answer to 2 decimal places.
Given further that \(0 < k < 10\)
(b) find the maximum temperature inside the greenhouse,
(c) find the time of day at which this maximum temperature occurs.
Give your answer to the nearest minute.