1.05o Trigonometric equations: solve in given intervals

1022 questions

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CAIE P3 2018 June Q10
8 marks Moderate -0.3
  1. Solve the equation \(2 \cos x + 3 \sin x = 0\), for \(0° \leqslant x \leqslant 360°\). [3]
  2. Sketch, on the same diagram, the graphs of \(y = 2 \cos x\) and \(y = -3 \sin x\) for \(0° \leqslant x \leqslant 360°\). [3]
  3. Use your answers to parts (i) and (ii) to find the set of values of \(x\) for \(0° \leqslant x \leqslant 360°\) for which \(2 \cos x + 3 \sin x > 0\). [2]
CAIE P3 2018 June Q7
9 marks Moderate -0.3
    1. Express \(\frac{\tan^2 \theta - 1}{\tan^2 \theta + 1}\) in the form \(a \sin^2 \theta + b\), where \(a\) and \(b\) are constants to be found. [3]
    2. Hence, or otherwise, and showing all necessary working, solve the equation $$\frac{\tan^2 \theta - 1}{\tan^2 \theta + 1} = \frac{1}{4}$$ for \(-90° \leqslant \theta \leqslant 0°\). [2]
  2. \includegraphics{figure_7b} The diagram shows the graphs of \(y = \sin x\) and \(y = 2 \cos x\) for \(-\pi \leqslant x \leqslant \pi\). The graphs intersect at the points \(A\) and \(B\).
    1. Find the \(x\)-coordinate of \(A\). [2]
    2. Find the \(y\)-coordinate of \(B\). [2]
CAIE P3 2018 November Q2
4 marks Moderate -0.3
Showing all necessary working, solve the equation \(\sin(\theta - 30°) + \cos \theta = 2 \sin \theta\), for \(0° < \theta < 180°\). [4]
CAIE P3 2018 November Q6
8 marks Standard +0.8
  1. Show that the equation \((\sqrt{2})\cos ec x + \cot x = \sqrt{3}\) can be expressed in the form \(R\sin(x - \alpha) = \sqrt{2}\), where \(R > 0\) and \(0° < \alpha < 90°\). [4]
  2. Hence solve the equation \((\sqrt{2})\cos ec x + \cot x = \sqrt{3}\), for \(0° < x < 180°\). [4]
Edexcel P2 2022 June Q5
6 marks Standard +0.3
In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Solve, for \(-180° < \theta \leq 180°\), the equation $$3\tan(\theta + 43°) = 2\cos(\theta + 43°)$$ [6]
Edexcel C2 Q4
7 marks Moderate -0.3
  1. Show that the equation $$5 \cos^2 x = 3(1 + \sin x)$$ can be written as $$5 \sin^2 x + 3 \sin x - 2 = 0.$$ [2]
  2. Hence solve, for \(0 \leq x < 360°\), the equation $$5 \cos^2 x = 3(1 + \sin x),$$ giving your answers to 1 decimal place where appropriate. [5]
Edexcel C2 Q5
8 marks Moderate -0.8
Solve, for \(0 \leq x \leq 180°\), the equation $$\sin(x + 10°) = \frac{\sqrt{3}}{2}.$$ [4]
  1. \(\cos 2x = -0.9\), giving your answers to 1 decimal place. [4]
Edexcel C2 Q8
9 marks Moderate -0.8
  1. Find all the values of \(\theta\), to 1 decimal place, in the interval \(0° \leq \theta \leq 360°\) for which \(5 \sin (\theta + 30°) = 3\). [4]
  2. Find all the values of \(\theta\), to 1 decimal place, in the interval \(0° \leq \theta \leq 360°\) for which \(\tan^2 \theta = 4\). [5]
Edexcel C2 Q9
10 marks Moderate -0.8
  1. Sketch, for \(0 \leq x \leq 2\pi\), the graph of \(y = \sin\left(x + \frac{\pi}{6}\right)\). [2]
  2. Write down the exact coordinates of the points where the graph meets the coordinate axes. [3]
  3. Solve, for \(0 \leq x \leq 2\pi\), the equation \(\sin\left(x + \frac{\pi}{6}\right) = 0.65\), giving your answers in radians to 2 decimal places. [5]
Edexcel C2 Q4
9 marks Moderate -0.8
  1. Show that the equation \(3 \sin^2 \theta - 2 \cos^2 \theta = 1\) can be written as \(5 \sin^2 \theta = 3\). [2]
  2. Hence solve, for \(0° \leq \theta \leq 360°\), the equation \(3 \sin^2 \theta - 2 \cos^2 \theta = 1\), giving your answer to 1 decimal place. [7]
Edexcel C2 2008 January Q4
9 marks Moderate -0.8
  1. Show that the equation $$3 \sin^2 \theta - 2 \cos^2 \theta = 1$$ can be written as $$5 \sin^2 \theta = 3.$$ [2]
  2. Hence solve, for \(0° \leq \theta < 360°\), the equation $$3 \sin^2 \theta - 2 \cos^2 \theta = 1,$$ giving your answers to 1 decimal place. [7]
Edexcel C2 Q8
10 marks Moderate -0.3
  1. Solve, for \(0 \leq x < 360°\), the equation \(\cos (x - 20°) = -0.437\), giving your answers to the nearest degree. [4]
  2. Find the exact values of \(\theta\) in the interval \(0 \leq \theta < 360°\) for which $$3 \tan \theta = 2 \cos \theta.$$ [6]
Edexcel C2 Q1
8 marks Moderate -0.8
Find all values of \(\theta\) in the interval \(0 \leq \theta < 360\) for which
  1. \(\cos (\theta + 75)° = 0\). [3]
  2. \(\sin 2\theta ° = 0.7\), giving your answers to one decimal place. [5]
Edexcel C2 Q6
6 marks Moderate -0.3
Given that \(2 \sin 2\theta = \cos 2\theta\),
  1. show that \(\tan 2\theta = 0.5\). [1]
  2. Hence find the values of \(\theta\), to one decimal place, in the interval \(0 \leq \theta < 360\) for which \(2 \sin 2\theta ° = \cos 2\theta °\). [5]
Edexcel C2 Q14
8 marks Standard +0.3
Find the values of \(\theta\), to 1 decimal place, in the interval \(-180 \leq \theta < 180\) for which $$2 \sin^2 \theta ° - 2 \sin \theta ° = \cos^2 \theta °.$$ [8]
Edexcel C2 Q24
10 marks Moderate -0.8
$$f(x) = 5 \sin 3x°, \quad 0 \leq x \leq 180.$$
  1. Sketch the graph of \(f(x)\), indicating the value of \(x\) at each point where the graph intersects the \(x\)-axis. [3]
  2. Write down the coordinates of all the maximum and minimum points of \(f(x)\). [3]
  3. Calculate the values of \(x\) for which \(f(x) = 2.5\) [4]
Edexcel C2 Q33
13 marks Moderate -0.3
  1. Solve, for \(0° < x < 180°\), the equation $$\sin (2x + 50°) = 0.6,$$ giving your answers to 1 decimal place. [7]
  2. In the triangle \(ABC\), \(AC = 18\) cm, \(\angle ABC = 60°\) and \(\sin A = \frac{1}{3}\).
    1. Use the sine rule to show that \(BC = 4\sqrt{3}\). [4]
    2. Find the exact value of \(\cos A\). [2]
Edexcel C2 Q36
8 marks Moderate -0.8
  1. Sketch, for \(0 \leq x \leq 360°\), the graph of \(y = \sin (x + 30°)\). [2]
  2. Write down the coordinates of the points at which the graph meets the axes. [3]
  3. Solve, for \(0 \leq x < 360°\), the equation $$\sin (x + 30°) = -\frac{1}{2}.$$ [3]
Edexcel C3 Q8
9 marks Standard +0.3
  1. Prove that $$\frac{1 - \cos 2\theta}{\sin 2\theta} = \tan \theta, \quad \theta \neq \frac{n\pi}{2}, \quad n \in \mathbb{Z}.$$ [3]
  2. Solve, giving exact answers in terms of \(\pi\), $$2(1 - \cos 2\theta) = \tan \theta, \quad 0 < \theta < \pi.$$ [6]
Edexcel C3 Q36
12 marks Standard +0.3
    1. Express \((12 \cos \theta - 5 \sin \theta)\) in the form \(R \cos (\theta + \alpha)\), where \(R > 0\) and \(0 < \alpha < 90°\). [4]
    2. Hence solve the equation $$12 \cos \theta - 5 \sin \theta = 4,$$ for \(0 < \theta < 90°\), giving your answer to 1 decimal place. [3]
  2. Solve $$8 \cot \theta - 3 \tan \theta = 2,$$ for \(0 < \theta < 90°\), giving your answer to 1 decimal place. [5]
Edexcel FP2 Q7
11 marks Standard +0.8
  1. Use de Moivre's theorem to show that $$\sin 5\theta = 16 \sin^5 \theta - 20 \sin^3 \theta + 5 \sin \theta$$ [5]
Hence, given also that \(\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta\),
  1. find all the solutions of $$\sin 5\theta = 5 \sin 3\theta$$ in the interval \(0 \leq \theta < 2\pi\). Give your answers to 3 decimal places. [6]
Edexcel M2 2013 June Q7
16 marks Standard +0.3
\includegraphics{figure_4} A small ball is projected from a fixed point \(O\) so as to hit a target \(T\) which is at a horizontal distance \(9a\) from \(O\) and at a height \(6a\) above the level of \(O\). The ball is projected with speed \(\sqrt{(27ag)}\) at an angle \(\theta\) to the horizontal, as shown in Figure 4. The ball is modelled as a particle moving freely under gravity.
  1. Show that tan\(^2 \theta - 6\) tan \(\theta + 5 = 0\) [7]
The two possible angles of projection are \(\theta_1\) and \(\theta_2\), where \(\theta_1 > \theta_2\).
  1. Find tan \(\theta_1\) and tan \(\theta_2\). [3]
The particle is projected at the larger angle \(\theta_1\).
  1. Show that the time of flight from \(O\) to \(T\) is \(\sqrt{\left(\frac{78a}{g}\right)}\). [3]
  2. Find the speed of the particle immediately before it hits \(T\). [3]
AQA C2 2009 June Q8
9 marks Moderate -0.3
  1. Given that \(\frac{\sin \theta - \cos \theta}{\cos \theta} = 4\), prove that \(\tan \theta = 5\). [2]
    1. Use an appropriate identity to show that the equation $$2 \cos^2 x - \sin x = 1$$ can be written as $$2 \sin^2 x + \sin x - 1 = 0$$ [2]
    2. Hence solve the equation $$2 \cos^2 x - \sin x = 1$$ giving all solutions in the interval \(0° \leq x \leq 360°\). [5]
Edexcel C2 Q4
8 marks Moderate -0.8
  1. Sketch, for \(0 \leq x \leq 360°\), the graph of \(y = \sin (x + 30°)\). [2]
  2. Write down the coordinates of the points at which the graph meets the axes. [3]
  3. Solve, for \(0 \leq x < 360°\), the equation $$\sin (x + 30°) = -\frac{1}{2}.$$ [3]
Edexcel C2 Q5
8 marks Moderate -0.8
Find all values of θ in the interval 0 ≤ θ < 360 for which
  1. cos (θ + 75)° = 0. [3]
  2. sin 2θ° = 0.7, giving your answers to one decimal place. [5]