1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs

182 questions

Sort by: Default | Easiest first | Hardest first
AQA Paper 3 2021 June Q1
1 marks Easy -2.0
The graph of \(y = \arccos x\) is shown. \includegraphics{figure_1} State the coordinates of the end point \(P\). Circle your answer. [1 mark] \((-\pi, 1)\) \quad \((-1, \pi)\) \quad \(\left(-\frac{\pi}{2}, 1\right)\) \quad \(\left(-1, \frac{\pi}{2}\right)\)
AQA Further Paper 2 2020 June Q7
5 marks Standard +0.8
The diagram shows part of the graph of \(y = \cos^{-1} x\) \includegraphics{figure_7} The finite region enclosed by the graph of \(y = \cos^{-1} x\), the \(y\)-axis, the \(x\)-axis and the line \(x = 0.8\) is rotated by \(2\pi\) radians about the \(x\)-axis. Use Simpson's rule with five ordinates to estimate the volume of the solid formed. Give your answer to four decimal places. [5 marks]
SPS SPS FM Pure 2021 May Q7
9 marks Standard +0.3
Given that \(y = \arcsin x\), \(-1 \leqslant x < 1\),
  1. show that \(\frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}}\). [3]
Given that \(f(x) = \frac{3x + 2}{\sqrt{4 - x^2}}\),
  1. show that the mean value of \(f(x)\) over the interval \([0, \sqrt{2}]\), is $$\frac{\pi\sqrt{2}}{4} + A\sqrt{2} - A,$$ where \(A\) is a constant to be determined. [6]
OCR H240/02 2018 December Q8
7 marks Challenging +1.8
Use a suitable trigonometric substitution to find \(\int \frac{x^2}{\sqrt{1-x^2}} \text{d}x\). [7]
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 9795/2 Specimen Q6
12 marks Challenging +1.8
A light elastic string of natural length \(2a\) and modulus of elasticity \(\lambda\) is stretched between two points \(A\) and \(B\), which are \(3a\) apart on a smooth horizontal table. A particle of mass \(m\) is attached to the mid-point of the string, pulled aside to \(A\) and released.
  1. Prove that, while one part of the string is taut and the other part is slack, the particle is describing simple harmonic motion. [2]
  2. Find the speed of the particle when the slack part of the string becomes taut. [2]
  3. Prove that the total time for the particle to reach the mid-point of the string for the first time is $$\sqrt{\frac{ma}{\lambda}} \left( \frac{\pi}{3} + \frac{1}{\sqrt{2}} \sin^{-1} \frac{1}{\sqrt{7}} \right).$$ [8]
Edexcel AEA 2011 June Q1
Standard +0.3
Solve for \(0 \leq \theta \leq 180°\) $$\tan(\theta + 35°) = \cot(\theta - 53°)$$ [Total 4 marks]