1.05l Double angle formulae: and compound angle formulae

575 questions

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OCR MEI C4 Q4
16 marks Standard +0.3
\includegraphics{figure_3} In a theme park ride, a capsule C moves in a vertical plane (see Fig. 8). With respect to the axes shown, the path of C is modelled by the parametric equations $$x = 10 \cos \theta + 5 \cos 2\theta, \quad y = 10 \sin \theta + 5 \sin 2\theta, \quad (0 \leqslant \theta < 2\pi),$$ where \(x\) and \(y\) are in metres.
  1. Show that \(\frac{\text{d}y}{\text{d}x} = -\frac{\cos \theta + \cos 2\theta}{\sin \theta + \sin 2\theta}\). Verify that \(\frac{\text{d}y}{\text{d}x} = 0\) when \(\theta = \frac{1}{3}\pi\). Hence find the exact coordinates of the highest point A on the path of C. [6]
  2. Express \(x^2 + y^2\) in terms of \(\theta\). Hence show that $$x^2 + y^2 = 125 + 100 \cos \theta.$$ [4]
  3. Using this result, or otherwise, find the greatest and least distances of C from O. [2]
You are given that, at the point B on the path vertically above O, $$2 \cos^2 \theta + 2 \cos \theta - 1 = 0.$$
  1. Using this result, and the result in part (ii), find the distance OB. Give your answer to 3 significant figures. [4]
OCR MEI C4 Q5
7 marks Standard +0.3
Show that \(\cot 2\theta = \frac{1 - \tan^2 \theta}{2 \tan \theta}\). Hence solve the equation $$\cot 2\theta = 1 + \tan \theta \quad \text{for } 0° < \theta < 360°.$$ [7]
AQA FP2 2011 June Q7
16 marks Challenging +1.3
    1. Use de Moivre's Theorem to show that $$\cos 5\theta = \cos^5 \theta - 10 \cos^3 \theta \sin^2 \theta + 5 \cos \theta \sin^4 \theta$$ and find a similar expression for \(\sin 5\theta\). [5 marks]
    2. Deduce that $$\tan 5\theta = \frac{\tan \theta(5 - 10 \tan^2 \theta + \tan^4 \theta)}{1 - 10 \tan^2 \theta + 5 \tan^4 \theta}$$ [3 marks]
  2. Explain why \(t = \tan \frac{\pi}{5}\) is a root of the equation $$t^4 - 10t^2 + 5 = 0$$ and write down the three other roots of this equation in trigonometrical form. [3 marks]
  3. Deduce that $$\tan \frac{\pi}{5} \tan \frac{2\pi}{5} = \sqrt{5}$$ [5 marks]
OCR FP3 Q4
9 marks Standard +0.8
  1. By expressing \(\cos \theta\) in terms of \(e^{i\theta}\) and \(e^{-i\theta}\), show that $$\cos^5 \theta \equiv \frac{1}{16}(\cos 5\theta + 5\cos 3\theta + 10\cos \theta).$$ [5]
  2. Hence solve the equation \(\cos 5\theta + 5\cos 3\theta + 9\cos \theta = 0\) for \(0 \leqslant \theta \leqslant \pi\). [4]
OCR FP3 Q8
12 marks Standard +0.8
  1. By expressing \(\sin \theta\) in terms of \(e^{i\theta}\) and \(e^{-i\theta}\), show that $$\sin^6 \theta \equiv \frac{1}{32}(\cos 6\theta - 6\cos 4\theta + 15\cos 2\theta - 10).$$ [5]
  2. Replace \(\theta\) by \(\left(\frac{1}{2}\pi - \theta\right)\) in the identity in part (i) to obtain a similar identity for \(\cos^6 \theta\). [3]
  3. Hence find the exact value of \(\int_0^{2\pi} \left(\sin^6 \theta - \cos^6 \theta\right) d\theta\). [4]
OCR FP3 2011 June Q8
11 marks Challenging +1.2
  1. Use de Moivre's theorem to express \(\cos 4\theta\) as a polynomial in \(\cos \theta\). [4]
  2. Hence prove that \(\cos 4\theta \cos 2\theta \equiv 16 \cos^6 \theta - 24 \cos^4 \theta + 10 \cos^2 \theta - 1\). [1]
  3. Use part (ii) to show that the only roots of the equation \(\cos 4\theta \cos 2\theta = 1\) are \(\theta = n\pi\), where \(n\) is an integer. [3]
  4. Show that \(\cos 4\theta \cos 2\theta = -1\) only when \(\cos \theta = 0\). [3]
Edexcel AEA 2002 June Q1
8 marks Challenging +1.8
Solve the following equation, for \(0 \leq x \leq \pi\), giving your answers in terms of \(\pi\). $$\sin 5x - \cos 5x = \cos x - \sin x.$$ [8]
Edexcel AEA 2004 June Q1
9 marks Challenging +1.8
Solve the equation \(\cos x + \sqrt{(1 - \frac{1}{2} \sin 2x)} = 0\), in the interval \(0° \leq x < 360°\). [9]
Edexcel AEA 2008 June Q3
12 marks Challenging +1.8
  1. Prove that \(\tan 15° = 2 - \sqrt{3}\) [4]
  2. Solve, for \(0 < \theta < 360°\), $$\sin(\theta + 60°) \sin(\theta - 60°) = (1 - \sqrt{3}) \cos^2 \theta$$ [8]
OCR H240/02 2020 November Q6
3 marks Standard +0.3
Prove that \(\sqrt{2} \cos(2\theta + 45°) = \cos^2 \theta - 2\sin \theta \cos \theta - \sin^2 \theta\), where \(\theta\) is measured in degrees. [3]
OCR H240/02 2023 June Q4
9 marks Standard +0.3
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_4}
  1. Determine the coordinates of \(B\). [6]
  2. Hence determine the exact value of \(\tan \alpha\). [3]
AQA Paper 1 2024 June Q15
6 marks Standard +0.3
  1. Show that the expression $$\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta$$ can be written as $$4 \cos \theta - \sec \theta$$ where \(\sin \theta \neq 0\) and \(\cos \theta \neq 0\) [4 marks]
  2. A student is attempting to solve the equation $$\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta = 3 \quad \text{for } 0° \leq \theta \leq 360°$$ They use the result from part (a), and write the following incorrect solution: \(\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta = 3\) Step 1: \(4 \cos \theta - \sec \theta = 3\) Step 2: \(4 \cos \theta - \frac{1}{\cos \theta} - 3 = 0\) Step 3: \(4 \cos^2 \theta - 3 \cos \theta - 1 = 0\) Step 4: \(\cos \theta = 1\) or \(\cos \theta = -0.25\) Step 5: \(\theta = 0°, 104.5°, 255.5°, 360°\)
    1. Explain why the student should reject one of their values for \(\cos \theta\) in Step 4. [1 mark]
    2. State the correct solutions to the equation $$\sin 2\theta \cosec \theta + \cos 2\theta \sec \theta = 3 \quad \text{for } 0° \leq \theta \leq 360°$$ [1 mark]
AQA Paper 2 2020 June Q17
6 marks Standard +0.3
A ball is projected forward from a fixed point, \(P\), on a horizontal surface with an initial speed \(u\text{ ms}^{-1}\), at an acute angle \(\theta\) above the horizontal. The ball needs to first land at a point at least \(d\) metres away from \(P\). You may assume the ball may be modelled as a particle and that air resistance may be ignored. Show that $$\sin 2\theta \geq \frac{dg}{u^2}$$ [6 marks]
AQA Paper 2 2024 June Q6
6 marks Standard +0.8
It is given that $$(2 \sin \theta + 3 \cos \theta)^2 + (6 \sin \theta - \cos \theta)^2 = 30$$ and that \(\theta\) is obtuse. Find the exact value of \(\sin \theta\). Fully justify your answer. [6 marks]
AQA Paper 3 2018 June Q8
9 marks Standard +0.3
  1. Prove the identity \(\frac{\sin 2x}{1 + \tan^2 x} = 2\sin x \cos^3 x\) [3 marks]
  2. Hence find \(\int \frac{4\sin 4\theta}{1 + \tan^2 2\theta} d\theta\) [6 marks]
AQA Paper 3 2020 June Q9
5 marks Standard +0.3
  1. For \(\cos \theta \neq 0\), prove that $$\cosec 2\theta + \cot 2\theta = \cot \theta$$ [4 marks]
  2. Explain why $$\cot \theta \neq \cosec 2\theta + \cot 2\theta$$ when \(\cos \theta = 0\) [1 mark]
WJEC Unit 1 2024 June Q2
3 marks Moderate -0.8
Find all values of \(\theta\) in the range \(0° < \theta < 180°\) that satisfy the equation $$2\sin 2\theta = 1.$$ [3]
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]
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 2019 June Q5
8 marks Standard +0.8
  1. Show that \(\sin \theta - \sin 3\theta\) can be expressed in the form \(a\cos b\theta \sin \theta\), where \(a\), \(b\) are integers whose values are to be determined. [3]
  2. Find the mean value of \(y = 2\cos 2\theta \sin \theta + 7\) between \(\theta = 1\) and \(\theta = 3\), giving your answer correct to two decimal places. [5]
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 Q3
9 marks Standard +0.8
  1. Given that \(z = \cos\theta + i\sin\theta\), use de Moivre's theorem to show that $$z^n + \frac{1}{z^n} = 2\cos n\theta .$$ [3]
  2. Express \(32\cos^6\theta\) in the form \(a\cos 6\theta + b\cos 4\theta + c\cos 2\theta + d\), where \(a\), \(b\), \(c\), \(d\) are integers whose values are to be determined. [6]
WJEC Further Unit 4 2023 June Q5
7 marks Standard +0.8
  1. Write down and simplify the Maclaurin series for \(\sin 2x\) as far as the term in \(x^5\). [2]
  2. Using your answer to part (a), determine the Maclaurin series for \(\cos^2 x\) as far as the term in \(x^4\). [5]