1.05o Trigonometric equations: solve in given intervals

1022 questions

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SPS SPS SM Pure 2023 June Q16
8 marks Standard +0.3
\includegraphics{figure_5} A horizontal path connects an island to the mainland. On a particular morning, the height of the sea relative to the path, \(H\) m, is modelled by the equation $$H = 0.8 + k \cos(30t - 70)°$$ where \(k\) is a constant and \(t\) is number of hours after midnight. Figure 5 shows a sketch of the graph of \(H\) against \(t\). Use the equation of the model to answer parts (a), (b) and (c).
  1. Find the time of day at which the height of the sea is at its maximum. [2] Given that the maximum height of the sea relative to the path is 2 m,
    1. find a complete equation for the model,
    2. state the minimum height of the sea relative to the path.
    [2] It is safe to use the path when the sea is 10 centimetres or more below the path.
  3. Find the times between which it is safe to use the path. (Solutions relying entirely on calculator technology are not acceptable.) [4]
SPS SPS SM Pure 2023 October Q4
12 marks Moderate -0.3
$$f(x) = 12 \cos x - 4 \sin x.$$ Given that \(f(x) = R \cos(x + \alpha)\), where \(R \geq 0\) and \(0 \leq \alpha \leq 90°\),
  1. find the value of \(R\) and the value of \(\alpha\). [4]
  2. Hence solve the equation $$12 \cos x - 4 \sin x = 7$$ for \(0 \leq x < 360°\), giving your answers to one decimal place. [5]
    1. Write down the minimum value of \(12 \cos x - 4 \sin x\). [1]
    2. Find, to 2 decimal places, the smallest positive value of \(x\) for which this minimum value occurs. [2]
SPS SPS SM Pure 2023 September Q9
6 marks Challenging +1.2
Solve the following trigonometric equation in the range given. $$4\tan^2\theta\cos\theta = 15, \quad 0 \leq \theta < 360°.$$ [6 marks]
SPS SPS FM 2024 October Q9
6 marks Standard +0.8
  1. Factorise \(8xy - 4x + 6y - 3\) into the form \((ax + b)(cy + d)\) where \(a, b, c\) and \(d\) are integers
  2. Hence, or otherwise, solve $$8\sin(x^2)\cos\left(e^{\frac{x}{3}}\right) - 4\sin(x^2) + 6\cos\left(e^{\frac{x}{3}}\right) - 3 = 0$$ where \(0° < x < 19°\), giving your answers to 1 decimal place.
[6 marks]
SPS SPS FM 2023 October Q2
6 marks Moderate -0.8
Solve each of the following equations, for \(0° < x < 360°\).
  1. \(\sin \frac{1}{2}x = 0.8\) [3]
  2. \(\sin x = 3 \cos x\) [3]
SPS SPS FM Pure 2023 September Q7
8 marks Standard +0.8
  1. Prove the identity \(\frac{\cos x}{\sec x + 1} + \frac{\cos x}{\sec x - 1} = 2\cot^2 x\) [3 marks]
  2. Hence, solve the equation $$\frac{\cos\left(2\theta + \frac{\pi}{3}\right)}{\sec\left(2\theta + \frac{\pi}{3}\right) + 1} = \cot\left(2\theta + \frac{\pi}{3}\right) - \frac{\cos\left(2\theta + \frac{\pi}{3}\right)}{\sec\left(2\theta + \frac{\pi}{3}\right) - 1}$$ in the interval \(0 \leq \theta \leq 2\pi\), giving your values of \(\theta\) to three significant figures where appropriate. [5 marks]
SPS SPS SM 2025 February Q7
8 marks Standard +0.3
  1. Show that the equation $$2 \sin x \tan x = \cos x + 5$$ can be expressed in the form $$3 \cos^2 x + 5 \cos x - 2 = 0.$$ [3]
  2. Hence solve the equation $$2 \sin 2\theta \tan 2\theta = \cos 2\theta + 5,$$ giving all values of \(\theta\) between \(0°\) and \(180°\), correct to \(1\) decimal place. [5]
SPS SPS FM Pure 2025 June Q6
9 marks Standard +0.3
  1. Prove that $$1 - \cos 2\theta = \tan \theta \sin 2\theta, \quad \theta \neq \frac{(2n + 1)\pi}{2}, \quad n \in \mathbb{Z}$$ [3]
  2. Hence solve, for \(-\frac{\pi}{2} < x < \frac{\pi}{2}\), the equation $$(\sec^2 x - 5)(1 - \cos 2x) = 3\tan^2 x \sin 2x$$ Give any non-exact answer to 3 decimal places where appropriate. [6]
SPS SPS FM 2026 November Q5
8 marks Standard +0.3
  1. Show that the equation $$4\cos\theta - 1 = 2\sin\theta\tan\theta$$ can be written in the form $$6\cos^2\theta - \cos\theta - 2 = 0$$ [4]
  2. Hence solve, for \(0 \leq x < 90°\) $$4\cos 3x - 1 = 2\sin 3x\tan 3x$$ giving your answers, where appropriate, to one decimal place. (Solutions based entirely on graphical or numerical methods are not acceptable.) [4]
OCR H240/02 2018 December Q4
10 marks Standard +0.3
In this question you must show detailed reasoning.
  1. Show that \(\cos A + \sin A \tan A = \sec A\). [3]
  2. Solve the equation \(\tan 2\theta = 3 \tan \theta\) for \(0° \leqslant \theta \leqslant 180°\). [7]
OCR H240/01 2017 Specimen Q8
6 marks Standard +0.3
  1. Show that \(\frac{2\tan\theta}{1 + \tan^2\theta} = \sin 2\theta\). [3]
  1. In this question you must show detailed reasoning. Solve \(\frac{2\tan\theta}{1 + \tan^2\theta} = 3\cos 2\theta\) for \(0 \leq \theta \leq \pi\). [3]
OCR H240/03 2017 Specimen Q3
4 marks Standard +0.8
In this question you must show detailed reasoning. Given that \(5\sin 2x = 3\cos x\), where \(0° < x < 90°\), find the exact value of \(\sin x\). [4]
OCR AS Pure 2017 Specimen Q2
5 marks Standard +0.3
In this question you must show detailed reasoning. Solve the equation \(2\cos^2 x = 2 - \sin x\) for \(0° \leq x \leq 180°\). [5]
Pre-U Pre-U 9794/2 2010 June Q4
6 marks Standard +0.3
  1. Show that $$\cos^4 x - \sin^4 x = 2\cos^2 x - 1.$$ [2]
  2. Hence find the solutions of $$\cos^4 x - \sin^4 x = \cos x,$$ where \(0° \leqslant x \leqslant 360°\). [4]
Pre-U Pre-U 9794/1 2011 June Q9
9 marks Standard +0.8
  1. Prove that \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\) and deduce that $$\sin \theta + \sin 3\theta = 4 \sin \theta \cos^2 \theta.$$ [5]
  2. Hence find the values of \(\theta\) such that \(0° < \theta < 180°\) that satisfy the equation $$\cot^2 \theta = \sin \theta + \sin 3\theta.$$ [4]
Pre-U Pre-U 9794/2 2011 June Q4
9 marks Standard +0.3
  1. On the same diagram, sketch the graphs of \(y = 2 \sec x\) and \(y = 1 + 3 \cos x\), for \(0 \leqslant x \leqslant \pi\). [4]
  2. Solve the equation \(2 \sec x = 1 + 3 \cos x\), where \(0 \leqslant x \leqslant \pi\). [5]
Pre-U Pre-U 9794/2 2011 June Q8
15 marks Challenging +1.3
  1. A curve \(C_1\) is defined by the parametric equations $$x = \theta - \sin \theta, \quad y = 1 - \cos \theta,$$ where the parameter \(\theta\) is measured in radians.
    1. Show that \(\frac{dy}{dx} = \cot \frac{1}{2}\theta\), except for certain values of \(\theta\), which should be identified. [5]
    2. Show that the points of intersection of the curve \(C_1\) and the line \(y = x\) are determined by an equation of the form \(\theta = 1 + A \sin(\theta - \alpha)\), where \(A\) and \(\alpha\) are constants to be found, such that \(A > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). [4]
    3. Show that the equation found in part (b) has a root between \(\frac{1}{4}\pi\) and \(\pi\). [2]
  2. A curve \(C_2\) is defined by the parametric equations $$x = \theta - \frac{1}{2} \sin \theta, \quad y = 1 - \frac{1}{2} \cos \theta,$$ where the parameter \(\theta\) is measured in radians. Find the y-coordinates of all points on \(C_2\) for which \(\frac{d^2y}{dx^2} = 0\). [4]
Pre-U Pre-U 9795/1 2015 June Q6
9 marks Challenging +1.2
  1. Given the complex number \(z = \cos \theta + \text{i} \sin \theta\), show that \(z^n + \frac{1}{z^n} = 2 \cos n\theta\). [1]
  2. Deduce the identity \(16 \cos^5 \theta \equiv \cos 5\theta + 5 \cos 3\theta + 10 \cos \theta\). [4]
  3. For \(0 < \theta < 2\pi\), solve the equation \(\cos 5\theta + 5 \cos 3\theta + 9 \cos \theta = 0\). [4]
Pre-U Pre-U 9795/1 2018 June Q9
8 marks Standard +0.3
  1. Use de Moivre's theorem to prove that \(\cos 3\theta = 4c^3 - 3c\), where \(c = \cos\theta\). [3]
  2. Solve the equation \(2\cos 3\theta - \sqrt{3} = 0\) for \(0 < \theta < \pi\), giving each answer in an exact form. [2]
  3. Deduce, in trigonometric form, the three roots of the equation \(x^3 - 3x - \sqrt{3} = 0\). [3]
Edexcel AEA 2014 June Q2
6 marks Challenging +1.2
Given that $$3\sin^2 x + 2\sin x = 6\cos x + 9\sin x \cos x$$ and that \(-90° < x < 90°\), find the possible values of \(\tan x\). [6]
Edexcel AEA 2011 June Q1
Standard +0.3
Solve for \(0 \leq \theta \leq 180°\) $$\tan(\theta + 35°) = \cot(\theta - 53°)$$ [Total 4 marks]
Edexcel AEA 2015 June Q3
9 marks Challenging +1.8
Solve for \(0 < x < 360°\) $$\cot 2x - \tan 78° = \frac{(\sec x)(\sec 78°)}{2}$$ where \(x\) is not an integer multiple of \(90°\) [9]