1.05o Trigonometric equations: solve in given intervals

1022 questions

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CAIE P1 2011 November Q5
7 marks Moderate -0.3
  1. Given that $$3\sin^2 x - 8\cos x - 7 = 0,$$ show that, for real values of \(x\), $$\cos x = -\frac{2}{3}.$$ [3]
  2. Hence solve the equation $$3\sin^2(\theta + 70°) - 8\cos(\theta + 70°) - 7 = 0$$ for \(0° \leqslant \theta \leqslant 180°\). [4]
CAIE P1 2014 November Q5
6 marks Moderate -0.3
  1. Show that the equation \(1 + \sin x \tan x = 5 \cos x\) can be expressed as $$6 \cos^2 x - \cos x - 1 = 0.$$ [3]
  2. Hence solve the equation \(1 + \sin x \tan x = 5 \cos x\) for \(0° \leqslant x \leqslant 180°\). [3]
CAIE P1 2014 November Q11
10 marks Moderate -0.3
The function \(f : x \mapsto 6 - 4\cos(\frac{1}{2}x)\) is defined for \(0 \leqslant x \leqslant 2\pi\).
  1. Find the exact value of \(x\) for which \(f(x) = 4\). [3]
  2. State the range of \(f\). [2]
  3. Sketch the graph of \(y = f(x)\). [2]
  4. Find an expression for \(f^{-1}(x)\). [3]
CAIE P1 2014 November Q2
6 marks Standard +0.3
\includegraphics{figure_2} In the diagram, \(OADC\) is a sector of a circle with centre \(O\) and radius 3 cm. \(AB\) and \(CB\) are tangents to the circle and angle \(ABC = \frac{1}{4}\pi\) radians. Find, giving your answer in terms of \(\sqrt{3}\) and \(\pi\),
  1. the perimeter of the shaded region, [3]
  2. the area of the shaded region. [3]
CAIE P1 2014 November Q5
7 marks Moderate -0.3
  1. Show that \(\sin^2 \theta - \cos^4 \theta = 2 \sin^2 \theta - 1\). [3]
  2. Hence solve the equation \(\sin^2 \theta - \cos^4 \theta = \frac{1}{2}\) for \(0° \leq \theta \leq 360°\). [4]
CAIE P1 2016 November Q3
4 marks Moderate -0.8
Showing all necessary working, solve the equation \(6\sin^2 x - 5\cos^2 x = 2\sin^2 x + \cos^2 x\) for \(0° \leq x \leq 360°\). [4]
CAIE P1 2018 November Q6
7 marks Standard +0.3
\includegraphics{figure_6} The diagram shows a triangle \(ABC\) in which \(BC = 20\) cm and angle \(ABC = 90°\). The perpendicular from \(B\) to \(AC\) meets \(AC\) at \(D\) and \(AD = 9\) cm. Angle \(BCA = \theta°\).
  1. By expressing the length of \(BD\) in terms of \(\theta\) in each of the triangles \(ABD\) and \(DBC\), show that \(20\sin^2 \theta = 9\cos \theta\). [4]
  2. Hence, showing all necessary working, calculate \(\theta\). [3]
CAIE P2 2024 June Q7
10 marks Standard +0.3
  1. Prove that \(2\sin\theta\cosec 2\theta \equiv \sec\theta\). [2]
  2. Solve the equation \(\tan^2\theta + 7\sin\theta\cosec 2\theta = 8\) for \(-\pi < \theta < \pi\). [5]
  3. Find \(\int 8\sin^2\frac{1}{2}x\cosec^2 x \, dx\). [3]
CAIE P2 2023 March Q2
5 marks Standard +0.8
Solve the equation \(\tan(\theta - 60°) = 3 \cot \theta\) for \(-90° < \theta < 90°\). [5]
CAIE P2 2024 March Q7
10 marks Standard +0.8
  1. Prove that $$\sin 2\theta (a \cot\theta + b \tan\theta) \equiv a + b + (a - b) \cos 2\theta,$$ where \(a\) and \(b\) are constants. [4]
  2. Find the exact value of \(\int_{\frac{\pi}{12}}^{\frac{\pi}{6}} \sin 2\theta (5 \cot\theta + 3 \tan\theta) \mathrm{d}\theta\). [3]
  3. Solve the equation \(\sin^2\alpha\left(2\cot\frac{1}{2}\alpha + 7\tan\frac{1}{2}\alpha\right) = 11\) for \(-\pi < \alpha < \pi\). [3]
CAIE P2 2024 November Q4
3 marks Moderate -0.3
  1. Solve the equation \(\text{p}(\cos ec^2 \theta) = 0\) for \(-90° < \theta < 90°\). [3]
CAIE P2 2024 November Q7
11 marks Standard +0.8
  1. Prove that \(\cos(\theta + 30°)\cos(\theta + 60°) = \frac{1}{4}\sqrt{3} - \frac{1}{2}\sin 2\theta\). [4]
  2. Solve the equation \(5\cos(2\alpha + 30°)\cos(2\alpha + 60°) = 1\) for \(0° < \alpha < 90°\). [4]
  3. Show that the exact value of \(\cos 20° \cos 50° + \cos 40° \cos 70°\) is \(\frac{1}{2}\sqrt{3}\). [3]
CAIE P2 2015 June Q6
9 marks Standard +0.3
\includegraphics{figure_6} The diagram shows part of the curve with equation $$y = 4\sin^2 x + 8\sin x + 3$$ and its point of intersection \(P\) with the \(x\)-axis.
  1. Find the exact \(x\)-coordinate of \(P\). [3]
  2. Show that the equation of the curve can be written $$y = 5 + 8\sin x - 2\cos 2x,$$ and use integration to find the exact area of the shaded region enclosed by the curve and the axes. [6]
CAIE P2 2003 November Q4
7 marks Moderate -0.3
  1. Express \(\cos \theta + (\sqrt{3}) \sin \theta\) in the form \(R \cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), giving the exact value of \(\alpha\). [3]
  2. Hence show that one solution of the equation $$\cos \theta + (\sqrt{3}) \sin \theta = \sqrt{2}$$ is \(\theta = \frac{7}{12}\pi\), and find the other solution in the interval \(0 < \theta < 2\pi\). [4]
CAIE P2 2016 November Q7
10 marks Standard +0.3
  1. Express \(\sin 2\theta (3 \sec \theta + 4 \cosec \theta)\) in the form \(a \sin \theta + b \cos \theta\), where \(a\) and \(b\) are integers. [3]
  2. Hence express \(\sin 2\theta (3 \sec \theta + 4 \cosec \theta)\) in the form \(R \sin(\theta + \alpha)\) where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  3. Using the result of part (ii), solve the equation \(\sin 2\theta (3 \sec \theta + 4 \cosec \theta) = 7\) for \(0° \leq \theta \leq 360°\). [4]
CAIE P2 2018 November Q3
5 marks Standard +0.3
Solve the equation \(\sec^2 \theta = 3 \cosec \theta\) for \(0° < \theta < 180°\). [5]
CAIE P3 2024 June Q7
3 marks Standard +0.3
  1. Hence solve the equation $$8 \cos^3 \theta + 54 \cos^2 \theta - 17 \cos \theta - 21 = 0,$$ for \(0° \leqslant \theta \leqslant 360°\). [3]
CAIE P3 2021 March Q3
6 marks Standard +0.3
By first expressing the equation \(\tan(x + 45°) = 2 \cot x + 1\) as a quadratic equation in \(\tan x\), solve the equation for \(0° < x < 180°\). [6]
CAIE P3 2021 March Q5
8 marks Moderate -0.3
  1. Express \(\sqrt{7} \sin x + 2 \cos x\) in the form \(R \sin(x + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). State the exact value of \(R\) and give \(\alpha\) correct to 2 decimal places. [3]
  2. Hence solve the equation \(\sqrt{7} \sin 2\theta + 2 \cos 2\theta = 1\), for \(0° < \theta < 180°\). [5]
CAIE P3 2024 November Q7
6 marks Standard +0.3
  1. Show that the equation \(\tan^3 x + 2 \tan 2x - \tan x = 0\) may be expressed as $$\tan^3 x - 2 \tan^2 x - 3 = 0$$ for \(\tan x \neq 0\). [3]
  2. Hence solve the equation \(\tan^3 2\theta + 2 \tan 4\theta - \tan 2\theta = 0\) for \(0 < \theta < \pi\). Give your answers in exact form. [3]
CAIE P3 2006 June Q4
7 marks Moderate -0.3
  1. Express \(7\cos \theta + 24\sin \theta\) in the form \(R\cos(\theta - \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\), giving the exact value of \(R\) and the value of \(\alpha\) correct to 2 decimal places. [3]
  2. Hence solve the equation $$7\cos \theta + 24\sin \theta = 15,$$ giving all solutions in the interval \(0° \leqslant \theta \leqslant 360°\). [4]
CAIE P3 2010 June Q3
5 marks Standard +0.3
Solve the equation $$\tan(45° - x) = 2\tan x,$$ giving all solutions in the interval \(0° < x < 180°\). [5]
CAIE P3 2013 June Q7
9 marks Standard +0.3
  1. By first expanding \(\cos(x + 45°)\), express \(\cos(x + 45°) - (\sqrt{2}) \sin x\) in the form \(R \cos(x + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). Give the value of \(R\) correct to 4 significant figures and the value of \(\alpha\) correct to 2 decimal places. [5]
  2. Hence solve the equation $$\cos(x + 45°) - (\sqrt{2}) \sin x = 2,$$ for \(0° < x < 360°\). [4]
CAIE P3 2014 June Q3
5 marks Standard +0.3
Solve the equation $$\cos(x + 30°) = 2\cos x,$$ giving all solutions in the interval \(-180° < x < 180°\). [5]
CAIE P3 2017 June Q3
5 marks Standard +0.8
  1. Express the equation \(\cot \theta - 2 \tan \theta = \sin 2\theta\) in the form \(a \cos^4 \theta + b \cos^2 \theta + c = 0\), where \(a\), \(b\) and \(c\) are constants to be determined. [3]
  2. Hence solve the equation \(\cot \theta - 2 \tan \theta = \sin 2\theta\) for \(90° < \theta < 180°\). [2]