2020 NJC J2 CT1 Q2

Timothy Gan

2020 NJC J2 CT1 Q2

A teacher is conducting a lesson on angles. She asks Bob to verify that the angle of elevation from a fixed point $P$ to a fixed point $A$ at the top of a building is $\theta$ radians where $0<\theta <\frac{\pi }{2}$. Instead of standing at point $P$, Bob stood at point $B$ to take the measurement, shown in the diagram below. 

2020 NJC J2 CT1 Q2

The angle $BAP$ is denoted by $\beta$. The height of the building $OA$ and distance $AP$ are fixed constants $h$ and $a$ respectively. $OPB$ forms a straight line. 


Show that $AB=\frac{a}{\cos \beta -k\sin \beta }$, where $k$ is a constant expressed in terms of $\theta $.



Given that $\beta$ is sufficiently small angle, show that

$AB\approx a\left( 1+m\beta +n{{\beta }^{2}} \right)$,

where $m$ and $n$ are constants, in terms of $k$, to be determined.



It is further given that $k=2$. Find the set of positive values of $\beta$ for the approximation found in part (ii) to be valid, giving your answers to 4 significant figures.


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Published: 9th June 2022

Written by

Timothy Gan

This is Tim. Tim loves to teach math. Tim seeks to improve his teaching incessantly! Help Tim by telling him how he can do better.

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