1.05o Trigonometric equations: solve in given intervals

1022 questions

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CAIE P2 2006 June Q2
5 marks Moderate -0.3
2
  1. Prove the identity $$\cos \left( x + 30 ^ { \circ } \right) + \sin \left( x + 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \cos x$$
  2. Hence solve the equation $$\cos \left( x + 30 ^ { \circ } \right) + \sin \left( x + 60 ^ { \circ } \right) = 1$$ for \(0 ^ { \circ } < x < 90 ^ { \circ }\).
CAIE P2 2008 June Q5
7 marks Moderate -0.3
5
  1. Express \(5 \cos \theta - \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$5 \cos \theta - \sin \theta = 4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2009 June Q5
6 marks Standard +0.3
5 Solve the equation \(\sec x = 4 - 2 \tan ^ { 2 } x\), giving all solutions in the interval \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P2 2010 June Q3
6 marks Moderate -0.3
3
  1. Show that the equation \(\tan \left( x + 45 ^ { \circ } \right) = 6 \tan x\) can be written in the form $$6 \tan ^ { 2 } x - 5 \tan x + 1 = 0$$
  2. Hence solve the equation \(\tan \left( x + 45 ^ { \circ } \right) = 6 \tan x\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P2 2010 June Q8
9 marks Standard +0.3
8
  1. Prove the identity $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \sin x$$
  2. Hence solve the equation $$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) = \frac { 1 } { 2 } \sec x$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P2 2011 June Q8
9 marks Standard +0.8
8
  1. Prove that \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) \equiv 4 \cos 2 \theta\).
  2. Hence
    1. solve for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\) the equation \(\sin ^ { 2 } 2 \theta \left( \operatorname { cosec } ^ { 2 } \theta - \sec ^ { 2 } \theta \right) = 3\),
    2. find the exact value of \(\operatorname { cosec } ^ { 2 } 15 ^ { \circ } - \sec ^ { 2 } 15 ^ { \circ }\).
CAIE P2 2012 June Q4
7 marks Standard +0.3
4
  1. Given that \(35 + \sec ^ { 2 } \theta = 12 \tan \theta\), find the value of \(\tan \theta\).
  2. Hence, showing the use of an appropriate formula in each case, find the exact value of
    1. \(\tan \left( \theta - 45 ^ { \circ } \right)\),
    2. \(\tan 2 \theta\).
CAIE P2 2012 June Q4
8 marks Standard +0.3
4
  1. Express \(9 \sin \theta - 12 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places. Hence
  2. solve the equation \(9 \sin \theta - 12 \cos \theta = 4\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\),
  3. state the largest value of \(k\) for which the equation \(9 \sin \theta - 12 \cos \theta = k\) has any solutions.
CAIE P2 2013 June Q7
10 marks Standard +0.3
7
  1. Express \(5 \sin 2 \theta + 2 \cos 2 \theta\) in the form \(R \sin ( 2 \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. Hence
  2. solve the equation $$5 \sin 2 \theta + 2 \cos 2 \theta = 4$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\),
  3. determine the least value of \(\frac { 1 } { ( 10 \sin 2 \theta + 4 \cos 2 \theta ) ^ { 2 } }\) as \(\theta\) varies.
CAIE P2 2013 June Q8
9 marks Standard +0.8
8
  1. Prove the identity $$\frac { 1 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } \equiv \operatorname { cosec } x$$
  2. Hence solve the equation $$\frac { 2 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } = 3 \cot ^ { 2 } x - 2$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
CAIE P2 2014 June Q2
4 marks Standard +0.3
2 Solve the equation \(3 \sin 2 \theta \tan \theta = 2\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2015 June Q6
10 marks Standard +0.3
6
  1. Prove that \(2 \operatorname { cosec } 2 \theta \tan \theta \equiv \sec ^ { 2 } \theta\).
  2. Hence
    1. solve the equation \(2 \operatorname { cosec } 2 \theta \tan \theta = 5\) for \(0 < \theta < \pi\),
    2. find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 6 } \pi } 2 \operatorname { cosec } 4 x \tan 2 x \mathrm {~d} x\).
CAIE P2 2016 June Q2
5 marks Standard +0.3
2 Solve the equation \(5 \tan 2 \theta = 4 \cot \theta\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2016 June Q3
5 marks Standard +0.3
3
  1. Solve the equation \(| 3 u + 1 | = | 2 u - 5 |\).
  2. Hence solve the equation \(| 3 \cot x + 1 | = | 2 \cot x - 5 |\) for \(0 < x < \frac { 1 } { 2 } \pi\), giving your answer correct to 3 significant figures.
CAIE P2 2016 June Q4
8 marks Standard +0.3
4
  1. Show that \(\sin \left( \theta + 60 ^ { \circ } \right) + \sin \left( \theta + 120 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \cos \theta\).
  2. Hence
    1. find the exact value of \(\sin 105 ^ { \circ } + \sin 165 ^ { \circ }\),
    2. solve the equation \(\sin \left( \theta + 60 ^ { \circ } \right) + \sin \left( \theta + 120 ^ { \circ } \right) = \sec \theta\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P2 2017 June Q5
7 marks Moderate -0.3
5
  1. Express \(2 \cos \theta + ( \sqrt { } 5 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\) where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(2 \cos \theta + ( \sqrt { } 5 ) \sin \theta = 1\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P2 2018 June Q7
11 marks Challenging +1.2
7
  1. Express \(5 \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\). Give the value of \(\alpha\) correct to 4 decimal places.
  2. Using your answer from part (i), solve the equation $$5 \cot \theta - 4 \operatorname { cosec } \theta = 2$$ for \(0 < \theta < 2 \pi\).
  3. Find \(\int \frac { 1 } { \left( 5 \cos \frac { 1 } { 2 } x - 2 \sin \frac { 1 } { 2 } x \right) ^ { 2 } } \mathrm {~d} x\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P2 2018 June Q7
10 marks Standard +0.8
7
  1. Show that \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) \equiv \sec ^ { 2 } x\).
  2. Solve the equation \(2 \operatorname { cosec } ^ { 2 } 2 x ( 1 - \cos 2 x ) = \tan x + 21\) for \(0 < x < \pi\), giving your answers correct to 3 significant figures.
  3. Find \(\int \left[ 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) - 2 \operatorname { cosec } ^ { 2 } ( 4 y + 2 ) \cos ( 4 y + 2 ) \right] \mathrm { d } y\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE P3 2003 June Q1
4 marks Moderate -0.3
1
  1. Show that the equation $$\sin \left( x - 60 ^ { \circ } \right) - \cos \left( 30 ^ { \circ } - x \right) = 1$$ can be written in the form \(\cos x = k\), where \(k\) is a constant.
  2. Hence solve the equation, for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2005 June Q6
8 marks Standard +0.3
6
  1. Prove the identity $$\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3$$
  2. Hence solve the equation $$\cos 4 \theta + 4 \cos 2 \theta = 2$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2008 June Q4
7 marks Standard +0.3
4
  1. Show that the equation \(\tan \left( 30 ^ { \circ } + \theta \right) = 2 \tan \left( 60 ^ { \circ } - \theta \right)\) can be written in the form $$\tan ^ { 2 } \theta + ( 6 \sqrt { } 3 ) \tan \theta - 5 = 0$$
  2. Hence, or otherwise, solve the equation $$\tan \left( 30 ^ { \circ } + \theta \right) = 2 \tan \left( 60 ^ { \circ } - \theta \right) ,$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P3 2009 June Q3
5 marks Standard +0.3
3
  1. Prove the identity \(\operatorname { cosec } 2 \theta + \cot 2 \theta \equiv \cot \theta\).
  2. Hence solve the equation \(\operatorname { cosec } 2 \theta + \cot 2 \theta = 2\), for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2010 June Q2
6 marks Standard +0.3
2 Solve the equation $$\sin \theta = 2 \cos 2 \theta + 1$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2010 June Q3
7 marks Standard +0.3
3 It is given that \(\cos a = \frac { 3 } { 5 }\), where \(0 ^ { \circ } < a < 90 ^ { \circ }\). Showing your working and without using a calculator to evaluate \(a\),
  1. find the exact value of \(\sin \left( a - 30 ^ { \circ } \right)\),
  2. find the exact value of \(\tan 2 a\), and hence find the exact value of \(\tan 3 a\).
CAIE P3 2011 June Q9
10 marks Standard +0.8
9
  1. Prove the identity \(\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3\).
  2. Hence
    1. solve the equation \(\cos 4 \theta + 4 \cos 2 \theta = 1\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\),
    2. find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { 4 } \theta \mathrm {~d} \theta\).