Multiple independent equations — includes show/prove component

Two or more separate trigonometric equations where at least one part requires showing that an equation can be rewritten (e.g. converting to a quadratic, proving an identity) before solving.

5 questions · Standard +0.3

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Edexcel C12 2019 January Q14
11 marks Standard +0.3
14. In this question solutions based entirely on graphical or numerical methods are not acceptable.
  1. Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), the equation $$\sin \left( x + 60 ^ { \circ } \right) = - 0.4$$ giving your answers, in degrees, to one decimal place.
  2. (a) Show that the equation $$2 \sin \theta \tan \theta - 3 = \cos \theta$$ can be written in the form $$3 \cos ^ { 2 } \theta + 3 \cos \theta - 2 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$2 \sin \theta \tan \theta - 3 = \cos \theta$$ showing each stage of your working and giving your answers, in degrees, to one decimal place.
Edexcel C12 2017 October Q12
11 marks Standard +0.3
12. (i) Solve, for \(0 < \theta \leqslant 360 ^ { \circ }\), $$3 \sin \left( \theta + 30 ^ { \circ } \right) = 2 \cos \left( \theta + 30 ^ { \circ } \right)$$ giving your answers, in degrees, to 2 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
(ii) (a) Given that $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ show that $$\tan ^ { 2 } x = k , \quad \text { where } k \text { is a constant. }$$ (b) Hence solve, for \(0 < x \leqslant 2 \pi\), $$\frac { \cos ^ { 2 } x + 2 \sin ^ { 2 } x } { 1 - \sin ^ { 2 } x } = 5$$ giving your answers, in radians, to 3 decimal places.
Edexcel C2 2013 June Q8
11 marks Standard +0.3
8. (i) Solve, for \(- 180 ^ { \circ } \leqslant x < 180 ^ { \circ }\), $$\tan \left( x - 40 ^ { \circ } \right) = 1.5$$ giving your answers to 1 decimal place.
(ii) (a) Show that the equation $$\sin \theta \tan \theta = 3 \cos \theta + 2$$ can be written in the form $$4 \cos ^ { 2 } \theta + 2 \cos \theta - 1 = 0$$ (b) Hence solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), $$\sin \theta \tan \theta = 3 \cos \theta + 2$$ showing each stage of your working.
AQA C2 2011 January Q9
10 marks Standard +0.3
9
  1. Solve the equation \(\tan x = - 3\) in the interval \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
    1. Given that $$7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6$$ show that $$\tan ^ { 2 } \theta + \tan \theta - 6 = 0$$
    2. Hence solve the equation \(7 \sin ^ { 2 } \theta + \sin \theta \cos \theta = 6\) in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\), giving your answers to the nearest degree.
      (4 marks)
AQA C4 2011 January Q6
10 marks Standard +0.3
6
    1. Given that \(\tan 2 x + \tan x = 0\), show that \(\tan x = 0\) or \(\tan ^ { 2 } x = 3\).
    2. Hence find all solutions of \(\tan 2 x + \tan x = 0\) in the interval \(0 ^ { \circ } < x < 180 ^ { \circ }\).
      (l mark)
    1. Given that \(\cos x \neq 0\), show that the equation $$\sin 2 x = \cos x \cos 2 x$$ can be written in the form $$2 \sin ^ { 2 } x + 2 \sin x - 1 = 0$$
    2. Show that all solutions of the equation \(2 \sin ^ { 2 } x + 2 \sin x - 1 = 0\) are given by \(\sin x = \frac { \sqrt { 3 } - 1 } { p }\), where \(p\) is an integer.