- Use the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) to prove that
$$\cot x + \tan \left( \frac { x } { 2 } \right) = \operatorname { cosec } x \quad x \neq n \pi , n \in \mathbb { Z }$$
(ii)
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1e5324f5-a9bc-4041-bfbb-cb940417ea63-08_389_455_573_877}
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\caption{Figure 1}
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An engineer models the vertical height above the ground of the tip of one blade of a wind turbine, shown in Figure 1. The ground is assumed to be horizontal.
The vertical height of the tip of the blade above the ground, \(H\) metres, at time \(x\) seconds after the wind turbine has reached its constant operating speed, is modelled by the equation
$$H = 90 - 30 \cos ( 120 x ) ^ { \circ } - 40 \sin ( 120 x ) ^ { \circ }$$
- Show that \(H = 60\) when \(x = 0\)
Using the substitution \(t = \tan ( 60 x ) ^ { \circ }\)
- show that equation (I) can be rewritten as
$$H = \frac { 120 t ^ { 2 } - 80 t + 60 } { 1 + t ^ { 2 } }$$
- Hence find, according to the model, the value of \(x\) when the tip of the blade is 100 m above the ground for the first time after the wind turbine has reached its constant operating speed.