1.05o Trigonometric equations: solve in given intervals

1022 questions

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WJEC Unit 3 Specimen Q13
12 marks Standard +0.3
  1. Solve the equation $$\operatorname{cosec}^2 x + \cot^2 x = 5$$ for \(0^{\circ} \leq x \leq 360^{\circ}\). [5]
    1. Express \(4\sin \theta + 3\cos \theta\) in the form \(R\sin(\theta + \alpha)\), where \(R > 0\) and \(0^{\circ} \leq \alpha \leq 90^{\circ}\). [4]
    2. Solve the equation $$4\sin \theta + 3\cos \theta = 2$$ for \(0^{\circ} \leq \theta \leq 360^{\circ}\), giving your answer correct to the nearest degree. [3]
WJEC Further Unit 4 2019 June Q2
9 marks Challenging +1.2
  1. Show that \(3\sin x + 4\cos x - 2\) can be written as \(\frac{6t + 2 - 6t^2}{1 + t^2}\), where \(t = \tan\left(\frac{x}{2}\right)\). [2]
  2. Hence, find the general solution of the equation \(3\sin x + 4\cos x - 2 = 3\). [7]
WJEC Further Unit 4 2022 June Q3
9 marks Standard +0.8
  1. By putting \(t = \tan\left(\frac{\theta}{2}\right)\), show that the equation $$4\sin\theta + 5\cos\theta = 3$$ can be written in the form $$4t^2 - 4t - 1 = 0.$$ [3]
  2. Hence find the general solution of the equation $$4\sin\theta + 5\cos\theta = 3.$$ [6]
WJEC Further Unit 4 2022 June Q6
6 marks Challenging +1.8
Solve the equation $$\cos 2\theta - \cos 4\theta = \sin 3\theta \quad \text{for} \quad 0 \leq \theta \leq \pi$$ [6]
WJEC Further Unit 4 2023 June Q12
6 marks Challenging +1.2
Find the general solution of the equation $$\cos 4\theta + \cos 2\theta = \cos\theta.$$ [6]
WJEC Further Unit 4 2024 June Q9
9 marks Challenging +1.8
Find the general solution of the equation $$\sin 6\theta + 2\cos^2\theta = 3\cos 2\theta - \sin 2\theta + 1.$$ [9]
WJEC Further Unit 4 Specimen Q5
8 marks Challenging +1.2
Find all the roots of the equation $$\cos \theta + \cos 3\theta + \cos 5\theta = 0$$ lying in the interval \([0, \pi]\). Give all the roots in radians in terms of \(\pi\). [8]
SPS SPS SM 2020 June Q7
9 marks Moderate -0.3
  1. Solve, for \(-90° \leqslant \theta < 270°\), the equation, $$\sin(2\theta + 10°) = -0.6$$ giving your answers to one decimal place. [5]
    1. A student's attempt at the question "Solve, for \(-90° < x < 90°\), the equation \(7\tan x = 8\sin x\)" is set out below. \begin{align} 7\tan x &= 8\sin x
      7 \times \frac{\sin x}{\cos x} &= 8\sin x
      7\sin x &= 8\sin x \cos x
      7 &= 8\cos x
      \cos x &= \frac{7}{8}
      x &= 29.0° \text{ (to 3 sf)} \end{align} Identify two mistakes made by this student, giving a brief explanation of each mistake. [2]
    2. Find the smallest positive solution to the equation $$7\tan(4\alpha + 199°) = 8\sin(4\alpha + 199°)$$ [2]
SPS SPS FM 2020 December Q1
4 marks Moderate -0.3
Solve \(2 \sin x = \tan x\) exactly, where \(-\frac{\pi}{2} < x < \frac{\pi}{2}\). [4]
SPS SPS FM 2021 March Q2
5 marks Standard +0.3
  1. Express \(2 \tan^2 \theta - \frac{1}{\cos \theta}\) in terms of \(\sec \theta\). [1]
  2. Hence solve, for \(0° < \theta < 360°\), the equation $$2 \tan^2 \theta - \frac{1}{\cos \theta} = 4.$$ [4]
SPS SPS FM 2021 March Q5
13 marks Standard +0.3
  1. Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos (\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). Give the values of \(R\) and \(\alpha\) to 3 significant figures. [3]
The temperature \(T °C\), of an unheated building is modelled using the equation $$T = 15 + 2 \cos \left( \frac{\pi t}{12} \right) + 5 \sin \left( \frac{\pi t}{12} \right), \quad 0 \leq t < 24,$$ where \(t\) hours is the number of hours after 1200.
  1. Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs. [4]
  2. Calculate, to the nearest half hour, the times when the temperature is predicted to be \(12 °C\). [6]
SPS SPS FM 2021 April Q2
4 marks Challenging +1.2
solve, for \(0° < \theta < 360°\), the equation $$2 \tan^2 \theta - \frac{1}{\cos \theta} = 4.$$ [4]
SPS SPS FM 2021 April Q5
13 marks Standard +0.3
  1. Express \(2 \cos \theta + 5 \sin \theta\) in the form \(R \cos (\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\) Give the values of \(R\) and \(\alpha\) to 3 significant figures. [3]
The temperature \(T\) °C, of an unheated building is modelled using the equation $$T = 15 + 2 \cos \left( \frac{\pi t}{12} \right) + 5\sin \left( \frac{\pi t}{12} \right), \quad 0 \leq t < 24,$$ where \(t\) hours is the number of hours after 1200.
  1. Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs. [4]
  2. Calculate, to the nearest half hour, the times when the temperature is predicted to be 12 °C. [6]
SPS SPS SM Pure 2021 June Q9
7 marks Standard +0.3
\includegraphics{figure_3} Figure 3 shows part of the curve with equation \(y = 3\cos x^2\). The point \(P(c, d)\) is a minimum point on the curve with \(c\) being the smallest negative value of \(x\) at which a minimum occurs.
  1. State the value of \(c\) and the value of \(d\). [1]
  2. State the coordinates of the point to which \(P\) is mapped by the transformation which transforms the curve with equation \(y = 3\cos x^2\) to the curve with equation
    1. \(y = 3\cos\left(\frac{x^2}{4}\right)\)
    2. \(y = 3\cos(x - 36)^2\)
    [2]
  3. Solve, for \(450° \leq \theta < 720°\), $$3\cos\theta = 8\tan\theta$$ giving your solution to one decimal place. [4]
SPS SPS SM Pure 2021 June Q12
5 marks Standard +0.3
Solve the equation $$\sin\theta\tan\theta + 2\sin\theta = 3\cos\theta \quad \text{where } \cos\theta \neq 0$$ Give all values of \(\theta\) to the nearest degree in the interval \(0° < \theta < 180°\) Fully justify your answer. [5 marks]
SPS SPS SM Pure 2021 May Q7
13 marks Challenging +1.2
It is given that there is exactly one value of \(x\), where \(0 < x < \pi\), that satisfies the equation $$3\tan 2x - 8\tan x = 4.$$
  1. Show that \(t = \sqrt[3]{\frac{1}{2} + \frac{1}{3}t - \frac{1}{3}t^2}\), where \(t = \tan x\). [3]
  2. Show by calculation that the value of \(t\) satisfying the equation in part (i) lies between 0.7 and 0.8. [2]
  3. Use an iterative process based on the equation in part (i) to find the value of \(t\) correct to 4 significant figures. Use a starting value of 0.75 and show the result of each iteration. [3]
  4. Solve the equation \(3\tan 4y - 8\tan 2y = 4\) for \(0 < y < \frac{1}{4}\pi\). [2]
SPS SPS SM Pure 2020 October Q7
7 marks Standard +0.3
  1. Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \sin(\theta + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Hence solve the equation \(24 \sin \theta + 7 \cos \theta = 12\) for \(0° < \theta < 360°\). [4]
SPS SPS SM Pure 2020 October Q10
7 marks Standard +0.3
  1. Prove that $$\cos^2(\theta + 45°) - \frac{1}{2}(\cos 2\theta - \sin 2\theta) \equiv \sin^2 \theta.$$ [4]
  2. Hence solve the equation $$6\cos^2(\frac{1}{2}\theta + 45°) - 3(\cos \theta - \sin \theta) = 2$$ for \(-90° < \theta < 90°\). [3]
SPS SPS SM 2022 February Q3
8 marks Moderate -0.3
Solve each of the following equations, for \(0° \leqslant x \leqslant 180°\).
  1. \(2\sin^2 x = 1 + \cos x\). [4]
  2. \(\sin 2x = -\cos 2x\). [4]
SPS SPS FM Pure 2022 June Q13
8 marks Standard +0.3
  1. Show that \(\sin(2\theta + \frac{1}{2}\pi) = \cos 2\theta\). [2]
  2. Hence solve the equation \(\sin 3\theta = \cos 2\theta\) for \(0 \leq \theta \leq 2\pi\). [6]
SPS SPS SM Pure 2022 June Q10
6 marks Moderate -0.8
  1. Sketch the graph of \(y = \cos x\) in the interval \(0 \leq x \leq 2\pi\). State the values of the intercepts with the coordinate axes. [2 marks]
    1. Given that $$\sin^2 \theta = \cos \theta(2 - \cos \theta)$$ prove that \(\cos \theta = \frac{1}{2}\). [2 marks]
    2. Hence solve the equation $$\sin^2 2x = \cos 2x(2 - \cos 2x)$$ in the interval \(0 \leq x \leq \pi\) [2 marks]
SPS SPS SM Mechanics 2022 February Q7
7 marks Standard +0.3
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
  1. Show that $$\frac{1 - \cos 2\theta}{\sin^2 2\theta} = k \sec^2 \theta \quad \theta = \frac{n\pi}{2} \quad n \in \mathbb{Z}$$ where \(k\) is a constant to be found. [3]
  2. Hence solve, for \(-\frac{\pi}{2} < x < \frac{\pi}{2}\) $$\frac{1 - \cos 2x}{\sin^2 2x} = (1 + 2\tan x)^2$$ Give your answers to 3 significant figures where appropriate. [4]
SPS SPS SM 2021 November Q6
7 marks Challenging +1.2
  1. Prove the following trigonometric identities. You must show all of your algebraic steps clearly. $$(\cos x + \sin x)(\cos x - \sec x) \equiv 2 \cot 2x$$ [3]
  2. Solve the following equation for \(x\) in the interval \(0 \leq x \leq \pi\). Giving your answers in terms of \(\pi\). $$\sin\left(2x + \frac{\pi}{6}\right) = \frac{1}{2}\sin\left(2x - \frac{\pi}{6}\right)$$ [4]
SPS SPS FM Pure 2023 June Q2
6 marks Moderate -0.3
In this question you must show detailed reasoning.
  1. Express \(8\cos x + 5\sin x\) in the form \(R\cos(x - \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). [3]
  2. Hence solve the equation \(8\cos x + 5\sin x = 6\) for \(0 \leqslant x < 2\pi\), giving your answers correct to 4 decimal places. [3]
SPS SPS SM Pure 2023 June Q12
6 marks Standard +0.3
  1. Solve, for \(-180° \leq x < 180°\), the equation $$3 \sin^2 x + \sin x + 8 = 9 \cos^2 x$$ giving your answers to 2 decimal places. [4]
  2. Hence find the smallest positive solution of the equation $$3\sin^2(2\theta - 30°) + \sin(2\theta - 30°) + 8 = 9 \cos^2(2\theta - 30°)$$ giving your answer to 2 decimal places. [2]