Year: 2002
Paper: 2
Question Number: 6
Course: LFM Pure
Section: Trigonometry 2
No solution available for this problem.
Difficulty Rating: 1600.0
Difficulty Comparisons: 0
Banger Rating: 1484.0
Banger Comparisons: 1
The lines $l_1$, $l_2$ and $l_3$ lie in an inclined plane $P$ and pass through
a common point $A$. The line $l_2$ is a
line of greatest slope in $P$. The line $l_1$ is perpendicular to $l_3$ and
makes an acute angle $\alpha$ with $l_2$.
The angles between the horizontal and
$l_1$, $l_2$ and $l_3$ are $\pi/6$, $\beta$ and $\pi/4$, respectively.
Show that $\cos\alpha\sin\beta = \frac12\,$
and find the value of $\sin\alpha \sin\beta\,$. Deduce that
$\beta = \pi/3\,$.
The lines $l_1$ and $l_3$ are rotated in $P$ about
$A$ so that $l_1$ and $l_3$ remain perpendicular to each other.
The new acute angle between
$l_1$ and $l_2$ is $\theta$. The new angles which $l_1$ and $l_3$
make with the horizontal are $\phi$ and $2\phi$, respectively. Show that
\[
\tan^2\theta = \frac{3+\sqrt{13}}2\;.
\]