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2011 Paper 2 Q4
D: 1600.0 B: 1470.8
trigonometry trig identity sin nθ quadratic in sin exact values sin 18° double angle formula product-to-sum

  1. Find all the values of \(\theta\), in the range \(0^\circ < \theta < 180^\circ\), for which \(\cos\theta=\sin 4\theta\). Hence show that \[ \sin 18^\circ = \frac14\left( \sqrt 5 -1\right). \]
  2. Given that \[ 4\sin^2 x + 1 = 4\sin^2 2x \,, \] find all possible values of \(\sin x\,\), giving your answers in the form \(p+q\sqrt5\) where \(p\) and \(q\) are rational numbers.
  3. Hence find two values of \(\alpha\) with \(0^\circ < \alpha < 90^\circ\) for which \[ \sin^23\alpha + \sin^25\alpha = \sin^2 6\alpha\,. \]

Solution:

  1. Note that \(\cos \theta = \sin (90^\circ - \theta)\) so \begin{align*} && \sin(90^\circ - \theta) &= \sin 4 \theta\\ && 90^\circ - \theta &= 4\theta +360^{\circ}k \\ && 90^\circ + \theta &= 4\theta +360^{\circ}k \\ \Rightarrow && 5\theta &= 90^\circ, 450^\circ, 810^\circ, \cdots \\ && 3 \theta &= 90^\circ, 450^\circ, \cdots \\ \Rightarrow && \theta &= 18^\circ, 90^\circ, 162^\circ, \ldots \\ && \theta &= 30^\circ, 150^\circ, \ldots \end{align*} Therefore \(\theta = 8^\circ, 30^\circ, 90^\circ, 150^\circ, 162^\circ\). Note also that: \begin{align*} && 0 &= \sin 4 \theta - \cos \theta \\ &&&= 2 \sin 2 \theta \cos 2 \theta- \cos \theta \\ &&&= 4 \sin \theta \cos \theta \cos 2 \theta - \cos \theta \\ &&&= \cos \theta \left (4 \sin \theta (1- 2\sin^2 \theta) - 1 \right) \\ &&&= \cos \theta \left (-8\sin^3 \theta +4\sin \theta - 1 \right) \\ &&&= \cos \theta (1 - 2 \sin \theta)(4 \sin^2 \theta+2\sin \theta -1)\\ \cos \theta = 0: && \theta &= 90^\circ \\ \sin \theta = \frac12: && \theta &= 30^{\circ} \\ && \theta &= \sin^{-1} \left ( \frac{-1\pm \sqrt5}{4} \right) \end{align*} Therefore \(\sin 18^{\circ} = \frac{\pm \sqrt{5}-1}{4}\), but since \(\sin 18^{\circ} > 0\) it must be the positive version.
  2. \(\,\) \begin{align*} && 4 \sin^2 x + 1 &= 4 \sin^2 2 x \\ &&&= 16 \sin^2 x \cos^2 x \\ &&&= 16 \sin^2 x (1- \sin^2 x) \\ \Rightarrow && 0 &= 16y^2 -12y+1 \\ \Rightarrow && \sin^2 x &= \frac{3\pm \sqrt5}{8} \\ &&&= \left ( \frac{1 \pm \sqrt5}{4} \right)^2 \\ \Rightarrow && \sin x &= \pm \frac{1 \pm \sqrt{5}}{4} \end{align*}
  3. \(\,\) \begin{align*} && \sin^2 x + \frac1{2^2} &= \sin^2 2x \end{align*} So if we can have \(\sin 5x = \pm \frac12\) and \(\sin 3x = \pm \frac{1 \pm \sqrt5}{4}\) then we are good, ie \begin{align*} && 5x &= 30^{\circ}, 150^{\circ}, 210^{\circ}, 330^{\circ}, 390^{\circ}, \cdots \\ \Rightarrow && x &= 6^{\circ}, 30^{\circ}, 42^{\circ}, 66^{\circ}, 78^{\circ} \\ \Rightarrow && 3x &= \boxed{18^{\circ}}, \cancel{90^{\circ}}, \boxed{126^{\circ}}, \boxed{198^{\circ}}, \boxed{78^{\circ}} \end{align*} So our solutions are \(x = 6^{\circ}, 42^{\circ}, 66^{\circ}, 78^{\circ}\) although it's interesting to note that \(x = 45^{\circ}\) is another solution

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