1.05o Trigonometric equations: solve in given intervals

1022 questions

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CAIE P3 2008 November Q6
8 marks Standard +0.3
6
  1. Express \(5 \sin x + 12 \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation $$5 \sin 2 \theta + 12 \cos 2 \theta = 11$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2010 November Q3
5 marks Moderate -0.3
3 Solve the equation $$\cos \left( \theta + 60 ^ { \circ } \right) = 2 \sin \theta$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P3 2010 November Q8
9 marks Standard +0.3
8
  1. Express \(( \sqrt { } 6 ) \cos \theta + ( \sqrt { } 10 ) \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence, in each of the following cases, find the smallest positive angle \(\theta\) which satisfies the equation
    1. \(( \sqrt { } 6 ) \cos \theta + ( \sqrt { } 10 ) \sin \theta = - 4\),
    2. \(( \sqrt { } 6 ) \cos \frac { 1 } { 2 } \theta + ( \sqrt { } 10 ) \sin \frac { 1 } { 2 } \theta = 3\).
CAIE P3 2011 November Q6
8 marks Moderate -0.3
6
  1. Express \(\cos x + 3 \sin x\) in the form \(R \cos ( x - \alpha )\), 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 \(\cos 2 \theta + 3 \sin 2 \theta = 2\), for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P3 2011 November Q3
7 marks Standard +0.3
3
  1. Express \(8 \cos \theta + 15 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). Give the value of \(\alpha\) correct to 2 decimal places.
  2. Hence solve the equation \(8 \cos \theta + 15 \sin \theta = 12\), giving all solutions in the interval \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P3 2012 November Q3
5 marks Standard +0.3
3 Solve the equation $$\sin \left( \theta + 45 ^ { \circ } \right) = 2 \cos \left( \theta - 30 ^ { \circ } \right)$$ giving all solutions in the interval \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2014 November Q8
9 marks Standard +0.3
8
  1. By first expanding \(\sin ( 2 \theta + \theta )\), show that $$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$
  2. Show that, after making the substitution \(x = \frac { 2 \sin \theta } { \sqrt { 3 } }\), the equation \(x ^ { 3 } - x + \frac { 1 } { 6 } \sqrt { } 3 = 0\) can be written in the form \(\sin 3 \theta = \frac { 3 } { 4 }\).
  3. Hence solve the equation $$x ^ { 3 } - x + \frac { 1 } { 6 } \sqrt { } 3 = 0$$ giving your answers correct to 3 significant figures.
CAIE P3 2015 November Q6
8 marks Standard +0.8
6 The angles \(A\) and \(B\) are such that $$\sin \left( A + 45 ^ { \circ } \right) = ( 2 \sqrt { } 2 ) \cos A \quad \text { and } \quad 4 \sec ^ { 2 } B + 5 = 12 \tan B$$ Without using a calculator, find the exact value of \(\tan ( A - B )\).
CAIE P3 2016 November Q3
5 marks Standard +0.3
3 Express the equation \(\sec \theta = 3 \cos \theta + \tan \theta\) as a quadratic equation in \(\sin \theta\). Hence solve this equation for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P3 2016 November Q3
6 marks Standard +0.3
3 Express the equation \(\cot 2 \theta = 1 + \tan \theta\) as a quadratic equation in \(\tan \theta\). Hence solve this equation for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2017 November Q3
5 marks Standard +0.8
3 By expressing the equation \(\tan \left( \theta + 60 ^ { \circ } \right) + \tan \left( \theta - 60 ^ { \circ } \right) = \cot \theta\) in terms of \(\tan \theta\) only, solve the equation for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P3 2018 November Q6
8 marks Challenging +1.2
6
  1. Show that the equation ( \(\sqrt { } 2\) ) \(\operatorname { cosec } x + \cot x = \sqrt { } 3\) can be expressed in the form \(R \sin ( x - \alpha ) = \sqrt { } 2\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Hence solve the equation \(( \sqrt { } 2 ) \operatorname { cosec } x + \cot x = \sqrt { } 3\), for \(0 ^ { \circ } < x < 180 ^ { \circ }\).
CAIE P3 2019 November Q9
10 marks Standard +0.3
9
  1. By first expanding \(\cos ( 2 x + x )\), show that \(\cos 3 x \equiv 4 \cos ^ { 3 } x - 3 \cos x\).
  2. Hence solve the equation \(\cos 3 x + 3 \cos x + 1 = 0\), for \(0 \leqslant x \leqslant \pi\).
  3. Find the exact value of \(\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 3 } x \mathrm {~d} x\).
CAIE P3 2019 November Q4
7 marks Moderate -0.3
4
  1. Express \(( \sqrt { } 6 ) \sin x + \cos x\) in the form \(R \sin ( x + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). State the exact value of \(R\) and give \(\alpha\) correct to 3 decimal places.
  2. Hence solve the equation \(( \sqrt { } 6 ) \sin 2 \theta + \cos 2 \theta = 2\), for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P3 2019 November Q4
7 marks Standard +0.8
4
  1. By first expanding \(\tan ( 2 x + x )\), show that the equation \(\tan 3 x = 3 \cot x\) can be written in the form \(\tan ^ { 4 } x - 12 \tan ^ { 2 } x + 3 = 0\).
  2. Hence solve the equation \(\tan 3 x = 3 \cot x\) for \(0 ^ { \circ } < x < 90 ^ { \circ }\).
CAIE P3 Specimen Q3
6 marks Standard +0.8
3 The angles \(\theta\) and \(\phi\) lie between \(0 ^ { \circ }\) and \(180 ^ { \circ }\), and are such that $$\tan ( \theta - \phi ) = 3 \quad \text { and } \quad \tan \theta + \tan \phi = 1$$ Find the possible values of \(\theta\) and \(\phi\).
CAIE P2 2019 June Q7
10 marks Standard +0.8
7
  1. Show that \(2 \operatorname { cosec } 2 \theta \cot \theta \equiv \operatorname { cosec } ^ { 2 } \theta\).
  2. Hence show that \(\operatorname { cosec } ^ { 2 } 15 ^ { \circ } \tan 15 ^ { \circ } = 4\).
  3. Solve the equation \(2 \operatorname { cosec } \phi \cot \frac { 1 } { 2 } \phi + \operatorname { cosec } \frac { 1 } { 2 } \phi = 12\) for \(- 360 ^ { \circ } < \phi < 360 ^ { \circ }\). Show all necessary working.
    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 2019 June Q7
11 marks Standard +0.8
7
    1. Express \(4 \sin \theta + 4 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
    2. Hence find the smallest positive value of \(\theta\) satisfying the equation \(4 \sin \theta + 4 \cos \theta = 5\).
  2. Solve the equation $$4 \cot 2 x = 5 + \tan x$$ for \(0 < x < \pi\), showing all necessary working and giving the answers correct to 2 decimal places.
    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 2017 March Q2
5 marks Standard +0.3
2
  1. Given that \(\tan 2 \theta \cot \theta = 8\), show that \(\tan ^ { 2 } \theta = \frac { 3 } { 4 }\).
  2. Hence solve the equation \(\tan 2 \theta \cot \theta = 8\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2019 March Q1
4 marks Standard +0.3
1 Solve the equation \(\sec ^ { 2 } \theta + \tan ^ { 2 } \theta = 5 \tan \theta + 4\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\). Show all necessary working.
CAIE P2 2002 November Q5
9 marks Standard +0.3
5 The angle \(x\), measured in degrees, satisfies the equation $$\cos \left( x - 30 ^ { \circ } \right) = 3 \sin \left( x - 60 ^ { \circ } \right)$$
  1. By expanding each side, show that the equation may be simplified to $$( 2 \sqrt { } 3 ) \cos x = \sin x$$
  2. Find the two possible values of \(x\) lying between \(0 ^ { \circ }\) and \(360 ^ { \circ }\).
  3. Find the exact value of \(\cos 2 x\), giving your answer as a fraction.
CAIE P2 2004 November Q3
4 marks Moderate -0.3
3 Find the values of \(x\) satisfying the equation $$3 \sin 2 x = \cos x$$ for \(0 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).
CAIE P2 2005 November Q3
7 marks Standard +0.3
3
  1. Express \(12 \cos \theta - 5 \sin \theta\) 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 $$12 \cos \theta - 5 \sin \theta = 10$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P2 2006 November Q4
7 marks Standard +0.3
4
  1. Prove the identity $$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) \equiv 2 \tan 2 x .$$
  2. Hence solve the equation $$\tan \left( x + 45 ^ { \circ } \right) - \tan \left( 45 ^ { \circ } - x \right) = 2 ,$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P2 2007 November Q6
7 marks Standard +0.3
6
  1. Express \(8 \sin \theta - 15 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), 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 $$8 \sin \theta - 15 \cos \theta = 14$$ giving all solutions in the interval \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).