1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs

199 questions

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AQA C3 2011 June Q4
12 marks Standard +0.3
    1. Solve the equation \(\cosec \theta = -4\) for \(0° < \theta < 360°\), giving your answers to the nearest 0.1°. [2]
    2. Solve the equation $$2\cot^2(2x + 30°) = 2 - 7\cosec(2x + 30°)$$ for \(0° < x < 180°\), giving your answers to the nearest 0.1°. [6]
  2. Describe a sequence of two geometrical transformations that maps the graph of \(y = \cosec x\) onto the graph of \(y = \cosec(2x + 30°)\). [4]
Edexcel C3 Q7
8 marks Standard +0.3
  1. Express \(\sin x + \sqrt{3} \cos x\) in the form \(R \sin (x + \alpha)\), where \(R > 0\) and \(0 < \alpha < 90°\). [4]
  2. Show that the equation \(\sec x + \sqrt{3} \cosec x = 4\) can be written in the form $$\sin x + \sqrt{3} \cos x = 2 \sin 2x.$$ [3]
  3. Deduce from parts (a) and (b) that \(\sec x + \sqrt{3} \cosec x = 4\) can be written in the form $$\sin 2x - \sin (x + 60°) = 0.$$ [1]
Edexcel C3 Q5
8 marks Moderate -0.3
  1. Given that \(\sin x = \frac{3}{5}\), use an appropriate double angle formula to find the exact value of \(\sec 2x\). [4]
  2. Prove that $$\cot 2x + \cosec 2x \equiv \cot x, \quad \left(x \neq \frac{n\pi}{2}, n \in \mathbb{Z}\right).$$ [4]
Edexcel C3 Q5
7 marks Standard +0.3
  1. Prove, by counter-example, that the statement "\(\sec(A + B) \equiv \sec A + \sec B\), for all \(A\) and \(B\)" is false [2]
  2. Prove that $$\tan \theta + \cot \theta = 2\cosec 2\theta, \quad \theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}.$$ [5]
OCR C3 Q5
7 marks Moderate -0.3
  1. Write down the identity expressing \(\sin 2\theta\) in terms of \(\sin \theta\) and \(\cos \theta\). [1]
  2. Given that \(\sin \alpha = \frac{1}{4}\) and \(\alpha\) is acute, show that \(\sin 2\alpha = \frac{1}{8}\sqrt{15}\). [3]
  3. Solve, for \(0° < \beta < 90°\), the equation \(5 \sin 2\beta \sec \beta = 3\). [3]
OCR C3 Q2
5 marks Moderate -0.3
It is given that \(\theta\) is the acute angle such that \(\sin \theta = \frac{12}{13}\). Find the exact value of
  1. \(\cot \theta\), [2]
  2. \(\cos 2\theta\). [3]
OCR C3 Q7
9 marks Moderate -0.3
  1. Sketch the graph of \(y = \sec x\) for \(0 \leq x \leq 2\pi\). [2]
  2. Solve the equation \(\sec x = 3\) for \(0 \leq x \leq 2\pi\), giving the roots correct to 3 significant figures. [3]
  3. Solve the equation \(\sec \theta = 5 \cos \theta\) for \(0 \leq \theta \leq 2\pi\), giving the roots correct to 3 significant figures. [4]
OCR C3 Q9
12 marks Standard +0.8
  1. Prove the identity $$\tan(\theta + 60°) \tan(\theta - 60°) \equiv \frac{\tan^2 \theta - 3}{1 - 3 \tan^2 \theta}.$$ [4]
  2. Solve, for \(0° < \theta < 180°\), the equation $$\tan(\theta + 60°) \tan(\theta - 60°) = 4 \sec^2 \theta - 3,$$ giving your answers correct to the nearest \(0.1°\). [5]
  3. Show that, for all values of the constant \(k\), the equation $$\tan(\theta + 60°) \tan(\theta - 60°) = k^2$$ has two roots in the interval \(0° < \theta < 180°\). [3]
OCR C3 Q3
7 marks Moderate -0.3
  1. Solve, for \(0° < \alpha < 180°\), the equation \(\sec \frac{1}{2}\alpha = 4\). [3]
  2. Solve, for \(0° < \beta < 180°\), the equation \(\tan \beta = 7 \cot \beta\). [4]
OCR C3 2010 January Q2
8 marks Standard +0.3
The angle \(\theta\) is such that \(0° < \theta < 90°\).
  1. Given that \(\theta\) satisfies the equation \(6 \sin 2\theta = 5 \cos \theta\), find the exact value of \(\sin \theta\). [3]
  2. Given instead that \(\theta\) satisfies the equation \(8 \cos \theta \cosec^2 \theta = 3\), find the exact value of \(\cos \theta\). [5]
OCR C3 2009 June Q1
3 marks Easy -1.8
\includegraphics{figure_1} Each diagram above shows part of a curve, the equation of which is one of the following: $$y = \sin^{-1} x, \quad y = \cos^{-1} x, \quad y = \tan^{-1} x, \quad y = \sec x, \quad y = \cosec x, \quad y = \cot x.$$ State which equation corresponds to
  1. Fig. 1, [1]
  2. Fig. 2, [1]
  3. Fig. 3. [1]
OCR C3 2010 June Q3
6 marks Standard +0.3
  1. Express the equation \(\cosec \theta(3 \cos 2\theta + 7) + 11 = 0\) in the form \(a \sin^2 \theta + b \sin \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants. [3]
  2. Hence solve, for \(-180° < \theta < 180°\), the equation \(\cosec \theta(3 \cos 2\theta + 7) + 11 = 0\). [3]
Edexcel C3 Q7
12 marks Standard +0.3
  1. Use the identity $$\cos (A + B) = \cos A \cos B - \sin A \sin B$$ to prove that $$\cos x \equiv 1 - 2 \sin^2 \frac{x}{2}.$$ [3]
  2. Prove that, for \(\sin x \neq 0\), $$\frac{1 - \cos x}{\sin x} \equiv \tan \frac{x}{2}.$$ [3]
  3. Find the values of \(x\) in the interval \(0 \leq x \leq 360^{\circ}\) for which $$\frac{1 - \cos x}{\sin x} = 2 \sec^2 \frac{x}{2} - 5,$$ giving your answers to 1 decimal place where appropriate. [6]
OCR C3 Q4
8 marks Standard +0.3
$$\text{f}(x) = x^2 + 5x - 2 \sec x, \quad x \in \mathbb{R}, \quad -\frac{\pi}{2} < x < \frac{\pi}{2}.$$
  1. Show that the equation \(\text{f}(x) = 0\) has a root, \(\alpha\), such that \(1 < \alpha < 1.5\) [2]
  2. Show that a suitable rearrangement of the equation \(\text{f}(x) = 0\) leads to the iterative formula $$x_{n+1} = \cos^{-1} \left( \frac{2}{x_n^2 + 5x_n} \right).$$ [3]
  3. Use the iterative formula in part (ii) with a starting value of 1.25 to find \(\alpha\) correct to 3 decimal places. You should show the result of each iteration. [3]
OCR C3 Q3
6 marks Standard +0.8
Find all values of \(\theta\) in the interval \(-180 < \theta < 180\) for which $$\tan^2 \theta^\circ + \sec \theta^\circ = 1.$$ [6]
OCR C3 Q5
7 marks Standard +0.3
  1. Prove, by counter-example, that the statement "\(\cosec \theta - \sin \theta > 0\) for all values of \(\theta\) in the interval \(0 < \theta < \pi\)" is false. [2]
  2. Find the values of \(\theta\) in the interval \(0 < \theta < \pi\) such that $$\cosec \theta - \sin \theta = 2,$$ giving your answers to 2 decimal places. [5]
OCR MEI C4 2011 June Q5
6 marks Standard +0.3
Solve the equation \(\cosec^2 \theta = 1 + 2 \cot \theta\), for \(-180° \leqslant \theta \leqslant 180°\). [6]
OCR MEI C4 2013 June Q2
7 marks Standard +0.3
Show that the equation \(\cos ec x + 5 \cot x = 3 \sin x\) may be rearranged as $$3 \cos^2 x + 5 \cos x - 2 = 0.$$ Hence solve the equation for \(0° \leq x \leq 360°\), giving your answers to 1 decimal place. [7]
AQA Paper 1 2019 June Q7
11 marks Standard +0.3
  1. By sketching the graphs of \(y = \frac{1}{x}\) and \(y = \sec 2x\) on the axes below, show that the equation $$\frac{1}{x} = \sec 2x$$ has exactly one solution for \(x > 0\) [3 marks] \includegraphics{figure_7a}
  2. By considering a suitable change of sign, show that the solution to the equation lies between 0.4 and 0.6 [2 marks]
  3. Show that the equation can be rearranged to give $$x = \frac{1}{2}\cos^{-1}x$$ [2 marks]
    1. Use the iterative formula $$x_{n+1} = \frac{1}{2}\cos^{-1}x_n$$ with \(x_1 = 0.4\), to find \(x_2\), \(x_3\) and \(x_4\), giving your answers to four decimal places. [2 marks]
    2. On the graph below, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x_2\), \(x_3\) and \(x_4\). [2 marks] \includegraphics{figure_7d}
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]
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]
SPS SPS SM Pure 2020 October Q8
12 marks Challenging +1.3
    1. Sketch the graph of \(y = \cos \sec x\) for \(0 < x < 4\pi\). [3]
    2. It is given that \(\cos \sec \alpha = \cos \sec \beta\), where \(\frac{1}{2}\pi < \alpha < \pi\) and \(2\pi < \beta < \frac{5}{2}\pi\). By using your sketch, or otherwise, express \(\beta\) in terms of \(\alpha\). [2]
    1. Write down the identity giving \(\tan 2\theta\) in terms of \(\tan \theta\). [1]
    2. Given that \(\cot \phi = 4\), find the exact value of \(\tan \phi \cot 2\phi \tan 4\phi\), showing all your working. [6]
SPS SPS SM Mechanics 2022 February Q11
6 marks Challenging +1.2
The curve \(C\) has parametric equations $$x = \sin 2\theta \quad y = \cos\text{ec}^3 \theta \quad 0 < \theta < \frac{\pi}{2}$$
  1. Find an expression for \(\frac{dy}{dx}\) in terms of \(\theta\) [3]
  2. Hence find the exact value of the gradient of the tangent to \(C\) at the point where \(y = 8\) [3]
Pre-U Pre-U 9794/2 2011 June Q4
9 marks Standard +0.3
  1. On the same diagram, sketch the graphs of \(y = 2 \sec x\) and \(y = 1 + 3 \cos x\), for \(0 \leqslant x \leqslant \pi\). [4]
  2. Solve the equation \(2 \sec x = 1 + 3 \cos x\), where \(0 \leqslant x \leqslant \pi\). [5]