GP with trigonometric terms

15. The first three terms of a geometric series where \(\theta\) is a constant are $$- 8 \sin \theta , \quad 3 - 2 \cos \theta \quad \text { and } \quad 4 \cot \theta$$ a. Show that \(4 \cos ^ { 2 } \theta + 20 \cos \theta + 9 = 0\) Given that \(\theta\) lies in the interval \(90 ^ { \circ } < \theta < 180 ^ { \circ }\),
b. Find the value of \(\theta\).
c. Hence prove that this series is convergent.
d. Find \(S _ { \infty }\), in the form \(a ( 1 - \sqrt { 3 } )\)

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.} Given that the first three terms of a geometric series are $$12 \cos \theta \quad 5 + 2 \sin \theta \quad \text { and } \quad 6 \tan \theta$$

1. show that $$4 \sin ^ { 2 } \theta - 52 \sin \theta + 25 = 0$$ Given that \(\theta\) is an obtuse angle measured in radians,
2. solve the equation in part (a) to find the exact value of \(\theta\)
3. show that the sum to infinity of the series can be expressed in the form $$k ( 1 - \sqrt { 3 } )$$ where \(k\) is a constant to be found.

12

1. A geometric sequence has first term 1 and common ratio \(\frac { 1 } { 2 }\) 12
   1. (i) Find the sum to infinity of the sequence.
      12
   2. (ii) Hence, or otherwise, evaluate $$\sum _ { n = 1 } ^ { \infty } \left( \sin 30 ^ { \circ } \right) ^ { n }$$ 12
   3. Find the smallest positive exact value of \(\theta\), in radians, which satisfies the equation $$\sum _ { n = 0 } ^ { \infty } ( \cos \theta ) ^ { n } = 2 - \sqrt { 2 }$$

The first, second and third terms of a geometric progression are \(\sin\theta\), \(\cos\theta\) and \(2 - \sin\theta\) respectively, where \(\theta\) radians is an acute angle.

1. Find the value of \(\theta\). [3]
2. Using this value of \(\theta\), find the sum of the first 10 terms of the progression. Give the answer in the form \(\frac{b}{\sqrt{c} - 1}\), where \(b\) and \(c\) are integers to be found. [3]

Show that $$\sum_{n=2}^{\infty} \left(\frac{1}{4}\right)^n \cos(180n)^{\circ} = \frac{9}{28}$$ [3]