1.05o Trigonometric equations: solve in given intervals

1022 questions

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CAIE P1 2013 November Q8
10 marks Moderate -0.3
8 A function f is defined by \(\mathrm { f } : x \mapsto 3 \cos x - 2\) for \(0 \leqslant x \leqslant 2 \pi\).
  1. Solve the equation \(\mathrm { f } ( x ) = 0\).
  2. Find the range of f .
  3. Sketch the graph of \(y = \mathrm { f } ( x )\). A function g is defined by \(\mathrm { g } : x \mapsto 3 \cos x - 2\) for \(0 \leqslant x \leqslant k\).
  4. State the maximum value of \(k\) for which g has an inverse.
  5. Obtain an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2013 November Q7
7 marks Moderate -0.3
7
  1. Find the possible values of \(x\) for which \(\sin ^ { - 1 } \left( x ^ { 2 } - 1 \right) = \frac { 1 } { 3 } \pi\), giving your answers correct to 3 decimal places.
  2. Solve the equation \(\sin \left( 2 \theta + \frac { 1 } { 3 } \pi \right) = \frac { 1 } { 2 }\) for \(0 \leqslant \theta \leqslant \pi\), giving \(\theta\) in terms of \(\pi\) in your answers.
CAIE P1 2014 November Q3
4 marks Standard +0.8
3 Solve the equation \(\frac { 13 \sin ^ { 2 } \theta } { 2 + \cos \theta } + \cos \theta = 2\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
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 Q4
7 marks Standard +0.3
4
  1. Prove the identity \(\left( \frac { 1 } { \sin x } - \frac { 1 } { \tan x } \right) ^ { 2 } \equiv \frac { 1 - \cos x } { 1 + \cos x }\).
  2. Hence solve the equation \(\left( \frac { 1 } { \sin x } - \frac { 1 } { \tan x } \right) ^ { 2 } = \frac { 2 } { 5 }\) for \(0 \leqslant x \leqslant 2 \pi\).
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 \(b\).
CAIE P1 2016 November Q6
6 marks Standard +0.3
6
  1. Show that \(\cos ^ { 4 } x \equiv 1 - 2 \sin ^ { 2 } x + \sin ^ { 4 } x\).
  2. Hence, or otherwise, solve the equation \(8 \sin ^ { 4 } x + \cos ^ { 4 } x = 2 \cos ^ { 2 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
CAIE P1 2016 November Q2
5 marks Moderate -0.8
2
  1. Express the equation \(\sin 2 x + 3 \cos 2 x = 3 ( \sin 2 x - \cos 2 x )\) in the form \(\tan 2 x = k\), where \(k\) is a constant.
  2. Hence solve the equation for \(- 90 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }\).
CAIE P1 2016 November Q10
11 marks Standard +0.3
10 A function f is defined by \(\mathrm { f } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant \pi\).
  1. Find the range of f .
  2. Sketch the graph of \(y = \mathrm { f } ( x )\).
  3. Solve the equation \(\mathrm { f } ( x ) = 6\), giving answers in terms of \(\pi\). The function g is defined by \(\mathrm { g } : x \mapsto 5 - 2 \sin 2 x\) for \(0 \leqslant x \leqslant k\), where \(k\) is a constant.
  4. State the largest value of \(k\) for which g has an inverse.
  5. For this value of \(k\), find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
CAIE P1 2017 November Q7
9 marks Standard +0.3
7
  1. \includegraphics[max width=\textwidth, alt={}, center]{5201a3d5-7733-4d10-9de5-0c2255e3ad60-12_499_568_267_826} The diagram shows part of the graph of \(y = a + b \sin x\). Find the values of the constants \(a\) and \(b\).
    1. Show that the equation $$( \sin \theta + 2 \cos \theta ) ( 1 + \sin \theta - \cos \theta ) = \sin \theta ( 1 + \cos \theta )$$ may be expressed as \(3 \cos ^ { 2 } \theta - 2 \cos \theta - 1 = 0\).
    2. Hence solve the equation $$( \sin \theta + 2 \cos \theta ) ( 1 + \sin \theta - \cos \theta ) = \sin \theta ( 1 + \cos \theta )$$ for \(- 180 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 2017 November Q5
7 marks Standard +0.3
5
  1. Show that the equation \(\cos 2 x \left( \tan ^ { 2 } 2 x + 3 \right) + 3 = 0\) can be expressed as $$2 \cos ^ { 2 } 2 x + 3 \cos 2 x + 1 = 0$$
  2. Hence solve the equation \(\cos 2 x \left( \tan ^ { 2 } 2 x + 3 \right) + 3 = 0\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2017 November Q5
7 marks Moderate -0.3
5
  1. Show that the equation \(\frac { \cos \theta + 4 } { \sin \theta + 1 } + 5 \sin \theta - 5 = 0\) may be expressed as \(5 \cos ^ { 2 } \theta - \cos \theta - 4 = 0\).
    [0pt] [3]
  2. Hence solve the equation \(\frac { \cos \theta + 4 } { \sin \theta + 1 } + 5 \sin \theta - 5 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2018 November Q5
6 marks Standard +0.3
5
  1. Show that the equation $$\frac { \cos \theta - 4 } { \sin \theta } - \frac { 4 \sin \theta } { 5 \cos \theta - 2 } = 0$$ may be expressed as \(9 \cos ^ { 2 } \theta - 22 \cos \theta + 4 = 0\).
  2. Hence solve the equation $$\frac { \cos \theta - 4 } { \sin \theta } - \frac { 4 \sin \theta } { 5 \cos \theta - 2 } = 0$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
CAIE P1 2018 November Q7
7 marks Standard +0.3
7
  1. Show that \(\frac { \tan \theta + 1 } { 1 + \cos \theta } + \frac { \tan \theta - 1 } { 1 - \cos \theta } \equiv \frac { 2 ( \tan \theta - \cos \theta ) } { \sin ^ { 2 } \theta }\).
  2. Hence, showing all necessary working, solve the equation $$\frac { \tan \theta + 1 } { 1 + \cos \theta } + \frac { \tan \theta - 1 } { 1 - \cos \theta } = 0$$ for \(0 ^ { \circ } < \theta < 90 ^ { \circ }\).
CAIE P1 2019 November Q5
7 marks Standard +0.8
5
  1. Given that \(4 \tan x + 3 \cos x + \frac { 1 } { \cos x } = 0\), show, without using a calculator, that \(\sin x = - \frac { 2 } { 3 }\).
  2. Hence, showing all necessary working, solve the equation $$4 \tan \left( 2 x - 20 ^ { \circ } \right) + 3 \cos \left( 2 x - 20 ^ { \circ } \right) + \frac { 1 } { \cos \left( 2 x - 20 ^ { \circ } \right) } = 0$$ for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
CAIE P1 2019 November Q6
7 marks Moderate -0.3
6
  1. Given that \(x > 0\), find the two smallest values of \(x\), in radians, for which \(3 \tan ( 2 x + 1 ) = 1\). Show all necessary working.
  2. The function f : \(x \mapsto 3 \cos ^ { 2 } x - 2 \sin ^ { 2 } x\) is defined for \(0 \leqslant x \leqslant \pi\).
    1. Express \(\mathrm { f } ( x )\) in the form \(a \cos ^ { 2 } x + b\), where \(a\) and \(b\) are constants.
    2. Find the range of \(f\).
CAIE P1 2019 November Q7
7 marks Moderate -0.3
7
  1. Show that the equation \(3 \cos ^ { 4 } \theta + 4 \sin ^ { 2 } \theta - 3 = 0\) can be expressed as \(3 x ^ { 2 } - 4 x + 1 = 0\), where \(x = \cos ^ { 2 } \theta\).
  2. Hence solve the equation \(3 \cos ^ { 4 } \theta + 4 \sin ^ { 2 } \theta - 3 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).
CAIE P1 Specimen 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 P2 2020 June Q6
10 marks Standard +0.8
6
  1. Prove that $$\sin 2 \theta ( \operatorname { cosec } \theta - \sec \theta ) \equiv \sqrt { 8 } \cos \left( \theta + \frac { 1 } { 4 } \pi \right)$$
  2. Solve the equation $$\sin 2 \theta ( \operatorname { cosec } \theta - \sec \theta ) = 1$$ for \(0 < \theta < \frac { 1 } { 2 } \pi\). Give the answer correct to 3 significant figures.
  3. Find \(\int \sin x \left( \operatorname { cosec } \frac { 1 } { 2 } x - \sec \frac { 1 } { 2 } x \right) \mathrm { d } x\).
CAIE P2 2020 June Q8
10 marks Standard +0.3
8
  1. Show that \(3 \sin 2 \theta \cot \theta \equiv 6 \cos ^ { 2 } \theta\).
  2. Solve the equation \(3 \sin 2 \theta \cot \theta = 5\) for \(0 < \theta < \pi\).
  3. Find the exact value of \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } 3 \sin x \cot \frac { 1 } { 2 } x \mathrm {~d} x\).
    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 2021 June Q2
6 marks Moderate -0.3
2 By first expanding \(\sin \left( \theta + 30 ^ { \circ } \right)\), solve the equation \(\sin \left( \theta + 30 ^ { \circ } \right) \operatorname { cosec } \theta = 2\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P2 2021 June Q1
5 marks Moderate -0.3
1
  1. Solve the equation \(\ln ( 2 + x ) - \ln x = 2 \ln 3\).
  2. Hence solve the equation \(\ln ( 2 + \cot y ) - \ln ( \cot y ) = 2 \ln 3\) for \(0 < y < \frac { 1 } { 2 } \pi\). Give your answer correct to 4 significant figures.
CAIE P2 2021 June Q3
6 marks Standard +0.3
3 Solve the equation \(\sin \left( 2 \theta + 30 ^ { \circ } \right) = 5 \cos \left( 2 \theta + 60 ^ { \circ } \right)\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2022 June Q2
6 marks Standard +0.3
2
  1. Express the equation \(7 \tan \theta + 4 \cot \theta - 13 \sec \theta = 0\) in terms of \(\sin \theta\) only.
  2. Hence solve the equation \(7 \tan \theta + 4 \cot \theta - 13 \sec \theta = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P2 2023 June Q7
11 marks Standard +0.3
7
  1. Express \(7 \cos \theta + 24 \sin \theta\) 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. Solve the equation \(7 \cos \theta + 24 \sin \theta = 18\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
  3. As \(\beta\) varies, the greatest possible value of $$\frac { 150 } { 7 \cos \frac { 1 } { 2 } \beta + 24 \sin \frac { 1 } { 2 } \beta + 50 }$$ is denoted by \(V\).
    Find the value of \(V\) and determine the smallest positive value of \(\beta\) (in degrees) for which the value of \(V\) occurs.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.