1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=1

710 questions

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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 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 Q3
6 marks Standard +0.3
3
  1. Show that \(( \sec x + \cos x ) ^ { 2 }\) can be expressed as \(\sec ^ { 2 } x + a + b \cos 2 x\), where \(a\) and \(b\) are constants to be determined.
  2. Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } ( \sec x + \cos x ) ^ { 2 } \mathrm {~d} x\).
CAIE P2 2021 June Q4
8 marks Moderate -0.8
4
  1. Find the exact value of \(\int _ { 0 } ^ { 2 } 6 \mathrm { e } ^ { 2 x + 1 } \mathrm {~d} x\).
  2. Find \(\int \left( \tan ^ { 2 } x + 4 \sin ^ { 2 } 2 x \right) \mathrm { d } x\).
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 2022 June Q8
9 marks Challenging +1.2
8
  1. Express \(3 \sin 2 \theta \sec \theta + 10 \cos \left( \theta - 30 ^ { \circ } \right)\) 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.
  2. Hence solve the equation \(3 \sin 4 \beta \sec 2 \beta + 10 \cos \left( 2 \beta - 30 ^ { \circ } \right) = 2\) for \(0 ^ { \circ } < \beta < 90 ^ { \circ }\).
    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 2023 June Q6
7 marks Standard +0.3
6 Show that \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } \left( 4 \cos ^ { 2 } 2 x + \frac { 1 } { \cos ^ { 2 } x } \right) \mathrm { d } x = \frac { 3 } { 4 } \sqrt { 3 } + \frac { 1 } { 6 } \pi - 1\).
CAIE P2 2023 June Q1
5 marks Standard +0.3
1 Solve the equation $$\sec ^ { 2 } \theta + 5 \tan ^ { 2 } \theta = 9 + 17 \sec \theta$$ for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).
CAIE P2 2024 June Q4
7 marks Standard +0.8
4
  1. Show that \(3 \tan 2 \theta + \tan \left( \theta + 45 ^ { \circ } \right) \equiv \frac { \tan ^ { 2 } \theta + 8 \tan \theta + 1 } { 1 - \tan ^ { 2 } \theta }\).
  2. Hence solve the equation \(3 \tan 2 \theta + \tan \left( \theta + 45 ^ { \circ } \right) = 4\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\).
CAIE P2 2024 June Q7
10 marks Standard +0.3
7
  1. Prove that \(2 \sin \theta \operatorname { cosec } 2 \theta \equiv \sec \theta\).
  2. Solve the equation \(\tan ^ { 2 } \theta + 7 \sin \theta \operatorname { cosec } 2 \theta = 8\) for \(- \pi < \theta < \pi\). \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-12_2725_37_136_2010}
  3. Find \(\int 8 \sin ^ { 2 } \frac { 1 } { 2 } x \operatorname { cosec } ^ { 2 } x \mathrm {~d} x\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown. \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-14_2715_35_143_2012}
CAIE P2 2021 March Q2
5 marks Standard +0.3
2 Solve the equation \(\sec ^ { 2 } \theta \cot \theta = 8\) for \(0 < \theta < \pi\).