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2015 Paper 2 Q2
D: 1600.0 B: 1484.0
triangle geometry sine rule double angle formula trigonometric identities perpendicular bisector angle trisection angle bisector midpoint

In the triangle \(ABC\), angle \(BAC = \alpha\) and angle \(CBA= 2\alpha\), where \(2\alpha\) is acute, and \(BC= x\). Show that \(AB = (3-4 \sin^2\alpha)x\). The point \(D\) is the midpoint of \(AB\) and the point \(E\) is the foot of the perpendicular from \(C\) to \(AB\). Find an expression for \(DE\) in terms of \(x\). The point \(F\) lies on the perpendicular bisector of \(AB\) and is a distance \(x\) from \(C\). The points \(F\) and \(B\) lie on the same side of the line through \(A\) and \(C\). Show that the line \(FC\) trisects the angle \(ACB\).

Solution:

TikZ diagram
Note that the sine rule gives us \begin{align*} && \frac{x}{\sin \alpha} &= \frac{AB}{\sin (180-3\alpha)} \\ \Rightarrow && AB &= x \frac{\sin 3\alpha}{\sin \alpha} \\ &&&= x \frac{\sin \alpha \cos 2\alpha + \cos \alpha \sin 2\alpha}{\sin \alpha} \\ &&&= x \left ( \frac{\sin \alpha (1-2\sin^2\alpha) + 2(1-\sin^2 \alpha)\sin \alpha}{\sin \alpha} \right) \\ &&&= x (3 - 4\sin^2 \alpha) \end{align*}
TikZ diagram
Note that \(AD = (\tfrac32 - 2 \sin^2\alpha)x\) and \(AE = (3-4\sin^2\alpha-\cos2\alpha)x\) so \begin{align*} DE &= (3-4\sin^2\alpha-\cos2\alpha)x - (\tfrac32 - 2 \sin^2\alpha)x \\ &= (\tfrac32 - 2\sin^2 \alpha - (1-2\sin^2 \alpha))x \\ &= \tfrac12x \end{align*}
TikZ diagram
Since \(DE = \frac12x\) if we drop the perpendicular from \(F\) to \(EC\) we have a \(30-60-90\) triangle. Therefore \(\angle FCE = 30^\circ\). Notice that \(\angle CEB = 90^{\circ} - 2\alpha\) and \(\angle ACB = 180^\circ - 3\alpha\), therefore \begin{align*} \angle ACF &= \angle ACB - \angle FCE - \angle ECB \\ &= (180^\circ - 3\alpha) - 30^\circ - (90^{\circ} - 2\alpha) \\ &= 60^\circ - \alpha \\ &= \frac13 \angle ACB \end{align*}

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2011 Paper 1 Q11
D: 1516.0 B: 1516.0
moments non-uniform bar string over peg equilibrium trigonometry sine rule inclination angle rigid body

A thin non-uniform bar \(AB\) of length \(7d\) has centre of mass at a point \(G\), where \(AG=3d\). A light inextensible string has one end attached to \(A\) and the other end attached to \(B\). The string is hung over a smooth peg \(P\) and the bar hangs freely in equilibrium with \(B\) lower than~\(A\). Show that \[ 3\sin\alpha = 4\sin\beta\,, \] where \(\alpha\) and \(\beta\) are the angles \(PAB\) and \(PBA\), respectively. Given that \(\cos\beta=\frac45\) and that \(\alpha\) is acute, find in terms of \(d\) the length of the string and show that the angle of inclination of the bar to the horizontal is \(\arctan \frac17\,\).

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2005 Paper 1 Q7
D: 1500.0 B: 1516.0
telescoping product product notation trigonometry simplification cotangent partial fractions sine rule compound angle formula

The notation \(\displaystyle \prod^n_{r=1} \f (r)\) denotes the product $\f (1) \times \f (2) \times \f(3) \times \cdots \times \f(n)$. %For example, \(\displaystyle \prod_{r=1}^4 r = 24\). %Simplify \(\displaystyle \prod^n_{r=1} \frac{\g (r) }{ \g (r-1) }\). %You may assume that \(\g (r) \neq 0\) for any integer \(0 \le r \le n \). Simplify the following products as far as possible:

  1. \(\displaystyle \prod^n_{r=1} \l \frac{r+ 1 }{ r } \r\,\);
  2. \(\displaystyle \prod^n_{r=2} \l \frac{r^2 -1}{r^2 } \r\,\);
  3. $\displaystyle \prod^n_{r=1} \l {\cos \frac{2\pi }{ n} + \sin \frac{2\pi}{ n} \cot \frac{\l 2r-1 \r \pi }{ n} }\r\,$, where \(n\) is even.

Solution:

  1. \(\,\) \begin{align*} \prod^n_{r=1} \left ( \frac{r+ 1 }{ r } \right) &= \frac{2}{1} \cdot \frac{3}{2} \cdot \frac{4}{3} \cdots \frac{n-1}{n-2} \cdot \frac{n}{n-1} \cdot \frac{n+1}{n} \\ &= \frac{n+1}{1} = n+1 \end{align*}
  2. \(\,\) \begin{align*} \prod^n_{r=2} \left ( \frac{r^2 -1}{r^2 } \right) &= \prod^n_{r=2} \left ( \frac{(r -1)(r+1)}{r^2 } \right) \\ &= \left ( \frac{1}{2} \cdot \frac{3}{2} \right) \cdot \left ( \frac{2}{3} \cdot \frac{4}{3} \right) \cdots \left ( \frac{r-1}{r} \cdot \frac{r+1}{r}\right) \cdots \frac{n-1}{n} \cdot \frac{n+1}{n} \\ &= \frac{1}{n} \cdot \frac{n+1}{2} \\ &= \frac{n+1}{2n} \end{align*}
  3. When \(n\) is odd, the product is undefined, since we have a \(\cot \pi\) lurking in there. \begin{align*} \prod^n_{r=1} \left ( {\cos \frac{2\pi }{ n} + \sin \frac{2\pi}{ n} \cot \frac{ (2r-1 ) \pi }{ n} } \right) &= \prod^n_{r=1} \left ( {\cos \frac{2\pi }{ n} + \sin \frac{2\pi}{ n} \frac{\cos \frac{ (2r-1 ) \pi }{ n}}{\sin\frac{ (2r-1 ) \pi }{ n}} } \right) \\ &= \prod^n_{r=1} \frac{1}{\sin\frac{ (2r-1 ) \pi }{ n}} \left ( {\cos \frac{2\pi }{ n} \sin\frac{ (2r-1 ) \pi }{ n} + \sin \frac{2\pi}{ n} \cos \frac{ (2r-1 ) \pi }{ n} } \right) \\ &= \prod^n_{r=1} \frac{1}{\sin\frac{ (2r-1 ) \pi }{ n}} \sin \left ( \frac{2\pi}{n} + \frac{(2r-1)\pi}{n} \right) \\ &= \prod^n_{r=1} \frac{1}{\sin\frac{ (2r-1 ) \pi }{ n}} \sin \left ( \frac{(2r+1)\pi}{n} \right) \\ &= \frac{\sin \frac{3\pi}{n}}{\sin \frac{\pi}{n}} \cdot \frac{\sin \frac{5\pi}{n}}{\sin \frac{3\pi}{n}} \cdots \frac{\sin \frac{(2n+1)\pi}{n}}{\sin \frac{(2n-1)\pi}{n}} \\ &= \frac{\sin \frac{(2n+1)\pi}{n}}{\sin \frac{\pi}{n}} \\ &= 1 \end{align*}

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1990 Paper 2 Q2
D: 1600.0 B: 1500.0
trigonometry product-to-sum formulas circle theorems Ptolemy's theorem inscribed angle theorem geometric proof sine rule cyclic quadrilateral

Prove that if \(A+B+C+D=\pi,\) then \[ \sin\left(A+B\right)\sin\left(A+D\right)-\sin B\sin D=\sin A\sin C. \] The points \(P,Q,R\) and \(S\) lie, in that order, on a circle of centre \(O\). Prove that \[ PQ\times RS+QR\times PS=PR\times QS. \]

Solution: \begin{align*} \sin(A+B)\sin(A+D) - \sin B \sin D &= \sin (A+B)\sin(\pi - B-C) - \sin B \sin (\pi - A - B - C) \\ &= \sin (A+B)\sin(B+C) - \sin B \sin(A+B+C) \\ &= \sin(A+B)\sin (B+C) - \sin B (\sin (A+B)\cos C +\cos(A+B) \sin C) \\ &= \sin(A+B)\cos B \sin C + \cos(A+B)\sin B \sin C \\ &= \sin A \sin C \cos^2 B + \cos A \sin B \cos B \sin C - \cos A \cos B \sin B \sin C + \sin A \sin^2 B \sin C \\ &= \sin A \sin C (\cos^2 B + \sin^2 B) \\ &= \sin A \sin C \end{align*}

TikZ diagram
Using the extended form of the sine rule: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\) where \(R\) is the circumradius, we have \begin{align*} PR \times QS &= 2R \sin (A+D) \times 2R \sin (A+B) \\ &= 4R^2 \l \sin A \sin C + \sin B \sin D \r \\ &= 2R \sin A \times 2R \sin C + 2R \sin B 2R \sin D \\ &= PS \times QR + PQ \times RS \end{align*} as required.

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