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

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WJEC Unit 1 2024 June Q15
7 marks Standard +0.8
The diagram shows a sketch of part of the curve with equation \(y = 2\sin x + 3\cos^2 x - 3\). The curve crosses the \(x\)-axis at the points O, A, B and C. \includegraphics{figure_15} Find the value of \(x\) at each of the points A, B and C. [7]
WJEC Unit 1 Specimen Q2
6 marks Standard +0.3
Find all values of \(\theta\) between \(0°\) and \(360°\) satisfying $$7 \sin^2 \theta + 1 = 3 \cos^2 \theta - \sin \theta.$$ [6]
WJEC Unit 3 2018 June Q4
5 marks Standard +0.8
Solve the equation $$2\tan^2\theta + 2\tan\theta - \sec^2\theta = 2,$$ for values of \(\theta\) between \(0°\) and \(360°\). [5]
WJEC Unit 3 2023 June Q6
15 marks Standard +0.3
  1. Using the trigonometric identity \(\cos(A + B) = \cos A \cos B - \sin A \sin B\), show that the exact value of \(\cos 75°\) is \(\frac{\sqrt{6} - \sqrt{2}}{4}\). [3]
  2. Solve the equation \(2\cot^2 x + \cosec x = 4\) for values of \(x\) between \(0°\) and \(360°\). [6]
    1. Express \(7\cos\theta - 24\sin\theta\) in the form \(R\cos(\theta + \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0° < \alpha < 90°\).
    2. Find all values of \(\theta\) in the range \(0° < \theta < 360°\) satisfying $$7\cos\theta - 24\sin\theta = 5.$$ [6]
WJEC Unit 3 2024 June Q2
11 marks Standard +0.3
  1. Find all values of \(\theta\) in the range \(0° < \theta < 360°\) satisfying $$3\cot\theta + 4\cosec^2\theta = 5.$$ [5]
  2. By writing \(24\cos x - 7\sin x\) in the form \(R\cos(x + \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0° < \alpha < 90°\), solve the equation $$24\cos x - 7\sin x = 16$$ for values of \(x\) between \(0°\) and \(360°\). [6]
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 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 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 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 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 FM Pure 2023 June Q14
7 marks Challenging +1.8
A curve \(C\) has equation $$x^3 + y^3 = 3xy + 48$$ Prove that \(C\) has two stationary points and find their coordinates. [7]
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]
SPS SPS SM Pure 2023 October Q3
12 marks Moderate -0.3
  1. Given that \(\cos A = \frac{3}{4}\), where \(270° < A < 360°\), find the exact value of \(\sin 2A\). [5]
    1. Show that \(\cos\left(2x + \frac{\pi}{3}\right) + \cos\left(2x - \frac{\pi}{3}\right) = \cos 2x\). [3] Given that $$y = 3\sin^2 x + \cos\left(2x + \frac{\pi}{3}\right) + \cos\left(2x - \frac{\pi}{3}\right),$$
    2. show that \(\frac{dy}{dx} = \sin 2x\). [4]
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 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 FM 2024 October Q7
9 marks Standard +0.8
The diagram shows part of the graph of \(y = x^2\). The normal to the curve at the point \(A(1, 1)\) meets the curve again at \(B\). Angle \(AOB\) is denoted by \(\alpha\). \includegraphics{figure_7}
  1. Determine the coordinates of \(B\). [6]
  2. Hence determine the exact value of \(\tan\alpha\). [3]
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 Q4
5 marks Standard +0.8
Given that $$y = \frac{3\sin \theta}{2\sin \theta + 2\cos \theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$ show that $$\frac{dy}{d\theta} = \frac{A}{1 + \sin 2\theta} \quad -\frac{\pi}{4} < \theta < \frac{3\pi}{4}$$ where \(A\) is a rational constant to be found. [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]