4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.
Maths \& Computer Science and Computer Science applicants should turn to page 14. [Figure][Figure] A triangle $A B C$ has sides $B C , C A$ and $A B$ of sides $a , b$ and $c$ respectively, and angles at $A , B$ and $C$ are $\alpha , \beta$ and $\gamma$ where $0 \leqslant \alpha , \beta , \gamma \leqslant \frac { 1 } { 2 } \pi$. (i) Show that the area of $A B C$ equals $\frac { 1 } { 2 } b c \sin \alpha$. Deduce the sine rule $$\frac { a } { \sin \alpha } = \frac { b } { \sin \beta } = \frac { c } { \sin \gamma } .$$ (ii) The points $P , Q$ and $R$ are respectively the feet of the perpeniculars from $A$ to $B C$, $B$ to $C A$, and $C$ to $A B$ as shown. Prove that $$\text { Area of } P Q R = \left( 1 - \cos ^ { 2 } \alpha - \cos ^ { 2 } \beta - \cos ^ { 2 } \gamma \right) \times ( \text { Area of } A B C ) .$$ (iii) For what triangles $A B C$, with angles $\alpha , \beta , \gamma$ as above, does the equation $$\cos ^ { 2 } \alpha + \cos ^ { 2 } \beta + \cos ^ { 2 } \gamma = 1$$ hold?
\section*{4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY.}
Maths \& Computer Science and Computer Science applicants should turn to page 14.\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-12_508_595_701_376}\\
\includegraphics[max width=\textwidth, alt={}, center]{d1f785ad-1c78-46e3-9768-075d08bf73c7-12_502_599_712_1087}
A triangle $A B C$ has sides $B C , C A$ and $A B$ of sides $a , b$ and $c$ respectively, and angles at $A , B$ and $C$ are $\alpha , \beta$ and $\gamma$ where $0 \leqslant \alpha , \beta , \gamma \leqslant \frac { 1 } { 2 } \pi$.\\
(i) Show that the area of $A B C$ equals $\frac { 1 } { 2 } b c \sin \alpha$.
Deduce the sine rule
$$\frac { a } { \sin \alpha } = \frac { b } { \sin \beta } = \frac { c } { \sin \gamma } .$$
(ii) The points $P , Q$ and $R$ are respectively the feet of the perpeniculars from $A$ to $B C$, $B$ to $C A$, and $C$ to $A B$ as shown.
Prove that
$$\text { Area of } P Q R = \left( 1 - \cos ^ { 2 } \alpha - \cos ^ { 2 } \beta - \cos ^ { 2 } \gamma \right) \times ( \text { Area of } A B C ) .$$
(iii) For what triangles $A B C$, with angles $\alpha , \beta , \gamma$ as above, does the equation
$$\cos ^ { 2 } \alpha + \cos ^ { 2 } \beta + \cos ^ { 2 } \gamma = 1$$
hold?