Show then solve: tan/sin/cos identity manipulation

A question is this type if and only if it asks (a) to show that an equation involving tanθ (written as sinθ/cosθ) combined with sinθ or cosθ can be rewritten as a quadratic in sinθ or cosθ, then (b) solve in a given interval.

12 questions · Moderate -0.1

1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals
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CAIE P1 2023 March Q7
8 marks Moderate -0.3
7
  1. By first obtaining a quadratic equation in \(\cos \theta\), solve the equation $$\tan \theta \sin \theta = 1$$ for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
  2. Show that \(\frac { \tan \theta } { \sin \theta } - \frac { \sin \theta } { \tan \theta } \equiv \tan \theta \sin \theta\).
CAIE P1 2023 November Q3
7 marks Moderate -0.3
3
  1. Show that the equation $$5 \cos \theta - \sin \theta \tan \theta + 1 = 0$$ may be expressed in the form \(a \cos ^ { 2 } \theta + b \cos \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants to be found.
  2. Hence solve the equation \(5 \cos \theta - \sin \theta \tan \theta + 1 = 0\) for \(0 < \theta < 2 \pi\).
CAIE P1 2020 Specimen Q7
6 marks Standard +0.3
7
  1. Show that the equation \(1 + \sin x \tan x = 5 \cos x\) can be expressed as $$6 \cos ^ { 2 } x - \cos x - 1 = 0$$
  2. Hence solve the equation \(1 + \sin x \tan x = 5 \cos x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2002 June Q2
5 marks Moderate -0.3
2
  1. Show that \(\sin x \tan x\) may be written as \(\frac { 1 - \cos ^ { 2 } x } { \cos x }\).
  2. Hence solve the equation \(2 \sin x \tan x = 3\), for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2005 June Q3
4 marks Moderate -0.8
3
  1. Show that the equation \(\sin \theta + \cos \theta = 2 ( \sin \theta - \cos \theta )\) can be expressed as \(\tan \theta = 3\).
  2. Hence solve the equation \(\sin \theta + \cos \theta = 2 ( \sin \theta - \cos \theta )\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2008 June Q2
5 marks Moderate -0.3
2
  1. Show that the equation \(2 \tan ^ { 2 } \theta \cos \theta = 3\) can be written in the form \(2 \cos ^ { 2 } \theta + 3 \cos \theta - 2 = 0\).
  2. Hence solve the equation \(2 \tan ^ { 2 } \theta \cos \theta = 3\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2002 November Q5
6 marks Moderate -0.3
5
  1. Show that the equation \(3 \tan \theta = 2 \cos \theta\) can be expressed as $$2 \sin ^ { 2 } \theta + 3 \sin \theta - 2 = 0$$
  2. Hence solve the equation \(3 \tan \theta = 2 \cos \theta\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2015 November Q4
6 marks Moderate -0.3
4
  1. Show that the equation \(\frac { 4 \cos \theta } { \tan \theta } + 15 = 0\) can be expressed as $$4 \sin ^ { 2 } \theta - 15 \sin \theta - 4 = 0$$
  2. Hence solve the equation \(\frac { 4 \cos \theta } { \tan \theta } + 15 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2015 November Q7
8 marks Moderate -0.3
7
  1. Show that the equation \(\frac { 1 } { \cos \theta } + 3 \sin \theta \tan \theta + 4 = 0\) can be expressed as $$3 \cos ^ { 2 } \theta - 4 \cos \theta - 4 = 0$$ and hence solve the equation \(\frac { 1 } { \cos \theta } + 3 \sin \theta \tan \theta + 4 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
  2. \includegraphics[max width=\textwidth, alt={}, center]{5c1ab2aa-3609-4245-b87a-98ecedc83a11-3_581_773_1400_721} The diagram shows part of the graph of \(y = a \cos x - b\), where \(a\) and \(b\) are constants. The graph crosses the \(x\)-axis at the point \(C \left( \cos ^ { - 1 } c , 0 \right)\) and the \(y\)-axis at the point \(D ( 0 , d )\). Find \(c\) and \(d\) in terms of \(a\) and \(b\).
OCR C2 2005 January Q5
8 marks Moderate -0.3
5
  1. Prove that the equation $$\sin \theta \tan \theta = \cos \theta + 1$$ can be expressed in the form $$2 \cos ^ { 2 } \theta + \cos \theta - 1 = 0$$
  2. Hence solve the equation $$\sin \theta \tan \theta = \cos \theta + 1$$ giving all values of \(\theta\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
OCR C2 Q5
8 marks Moderate -0.3
  1. (i) Given that
$$8 \tan x - 3 \cos x = 0$$ show that $$3 \sin ^ { 2 } x + 8 \sin x - 3 = 0$$ (ii) Find, to 2 decimal places, the values of \(x\) in the interval \(0 \leq x \leq 2 \pi\) such that $$8 \tan x - 3 \cos x = 0$$
Edexcel AEA 2018 June Q2
7 marks Challenging +1.8
2.Solve,for \(0 \leqslant x \leqslant 360 ^ { \circ }\) $$\sin 47 ^ { \circ } \cos ^ { 3 } x + \cos 47 ^ { \circ } \sin x \cos ^ { 2 } x = \frac { 1 } { 2 } \cos ^ { 2 } x$$