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

710 questions

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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 9794/1 2010 June Q11
11 marks Challenging +1.2
  1. Write down an identity for \(\tan 2\theta\) in terms of \(\tan \theta\) and use this result to show that $$\tan 3\theta = \frac{3 \tan \theta - \tan^3 \theta}{1 - 3 \tan^2 \theta}.$$ [4]
  2. Given that \(0 < \theta < \frac{1}{2}\pi\) and \(\theta = \sin^{-1}\left(\frac{1}{\sqrt{10}}\right)\), show that \(\tan 3\theta = \frac{13}{3}\). [3]
  3. Show that the solutions of the equation $$\tan(3 \sin^{-1} x) = \frac{13}{3}$$ for \(0 < x < 2\pi\) are $$x = \frac{\sqrt{10}}{10} \quad \text{and} \quad x = \frac{\sqrt{10(1 + 3\sqrt{3})}}{20}.$$ [4]
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 2016 June Q9
11 marks Challenging +1.2
  1. Show that \(\frac{\sin x}{1 + \sin x} \equiv \sec x \tan x - \sec^2 x + 1\). [5]
  2. Hence show that \(\int_0^{\frac{\pi}{4}} \frac{\sin x}{1 + \sin x} \, dx = \frac{1}{4}\pi + \sqrt{2} - 2\). [6]
Pre-U Pre-U 9795/2 Specimen Q5
8 marks Moderate -0.3
A girl can paddle her canoe at \(5 \text{ m s}^{-1}\) in still water. She wishes to cross a river which is \(100 \text{ m}\) wide and flowing at \(8 \text{ m s}^{-1}\).
    1. Write down the angle to the river bank at which the boat must head, in order to cross the river in the least possible time. [1]
    2. Find the acute angle to the river bank at which the boat must head, in order to cross the river by the shortest route. [4]
  2. Calculate the times taken for each of the two cases in part (i). [3]
Pre-U Pre-U 9794/2 Specimen Q7
12 marks Moderate -0.3
  1. Given that \(\cos \theta = \frac{7}{25}\), where \(\frac{3}{2}\pi < \theta < 2\pi\), determine the exact values of
    1. \(\sin \theta\), [3]
    2. \(\sin(\frac{1}{2}\theta)\), [3]
    3. \(\sec(\frac{1}{2}\theta)\). [1]
    1. Express \(4 \cos x - 3 \sin x\) in the form \(A \cos(x + \alpha)\), where \(A > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). [2]
    2. Hence find the greatest and least values of \(4 \cos x - 3 \sin x\) for \(0 \leqslant x \leqslant \pi\). [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 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]