Trigonometric arithmetic progression

8 The first term of a progression is \(\sin ^ { 2 } \theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\). The second term of the progression is \(\sin ^ { 2 } \theta \cos ^ { 2 } \theta\).

1. Given that the progression is geometric, find the sum to infinity.
   It is now given instead that the progression is arithmetic.
2. 1. Find the common difference of the progression in terms of \(\sin \theta\).
   2. Find the sum of the first 16 terms when \(\theta = \frac { 1 } { 3 } \pi\).

9 The first term of a progression is \(\cos \theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\).

1. For the case where the progression is geometric, the sum to infinity is \(\frac { 1 } { \cos \theta }\).
   1. Show that the second term is \(\cos \theta \sin ^ { 2 } \theta\).
   2. Find the sum of the first 12 terms when \(\theta = \frac { 1 } { 3 } \pi\), giving your answer correct to 4 significant figures.
2. For the case where the progression is arithmetic, the first two terms are again \(\cos \theta\) and \(\cos \theta \sin ^ { 2 } \theta\) respectively. Find the 85 th term when \(\theta = \frac { 1 } { 3 } \pi\). \includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-16_547_421_264_863} The diagram shows a sector \(A B C\) which is part of a circle of radius \(a\). The points \(D\) and \(E\) lie on \(A B\) and \(A C\) respectively and are such that \(A D = A E = k a\), where \(k < 1\). The line \(D E\) divides the sector into two regions which are equal in area.

7 The first and second terms of an arithmetic progression are \(\frac { 1 } { \cos ^ { 2 } \theta }\) and \(- \frac { \tan ^ { 2 } \theta } { \cos ^ { 2 } \theta }\), respectively, where \(0 < \theta < \frac { 1 } { 2 } \pi\).

1. Show that the common difference is \(- \frac { 1 } { \cos ^ { 4 } \theta }\).
2. Find the exact value of the 13th term when \(\theta = \frac { 1 } { 6 } \pi\).

5 The first, third and fifth terms of an arithmetic progression are \(2 \cos x , - 6 \sqrt { 3 } \sin x\) and \(10 \cos x\) respectively, where \(\frac { 1 } { 2 } \pi < x < \pi\).

1. Find the exact value of \(x\).
2. Hence find the exact sum of the first 25 terms of the progression.

6 The first, third and fourth terms of an arithmetic progression are \(u _ { 1 } , u _ { 3 }\) and \(u _ { 4 }\) respectively, where \(u _ { 1 } = 2 \sin \theta , \quad u _ { 3 } = - \sqrt { 3 } \cos \theta , \quad u _ { 4 } = \frac { 7 } { 2 } \sin \theta\), and \(\frac { 1 } { 2 } \pi < \theta < \pi\).

1. Determine the exact value of \(\theta\).
2. Hence determine the value of \(\sum _ { r = 1 } ^ { 100 } u _ { r }\).

9

1. Three consecutive terms in an arithmetic sequence are \(3 \mathrm { e } ^ { - p } , 5,3 \mathrm { e } ^ { p }\) Find the possible values of \(p\). Give your answers in an exact form.
   [0pt] [6 marks]
   9
2. Prove that there is no possible value of \(q\) for which \(3 \mathrm { e } ^ { - q } , 5,3 \mathrm { e } ^ { q }\) are consecutive terms of a geometric sequence.
   [0pt] [4 marks]