1.05o Trigonometric equations: solve in given intervals

1022 questions

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Edexcel C2 Q7
9 marks Moderate -0.8
The curve C has equation \(y = \cos \left(x + \frac{\pi}{4}\right)\), \(0 \leq x \leq 2\pi\).
  1. Sketch C. [2]
  2. Write down the exact coordinates of the points at which C meets the coordinate axes. [3]
  1. Solve, for x in the interval \(0 \leq x \leq 2\pi\), $$\cos \left(x + \frac{\pi}{4}\right) = 0.5,$$ giving your answers in terms of π. [4]
Edexcel C2 Q7
14 marks Standard +0.3
Find all the values of \(\theta\) in the interval \(0 \leq \theta < 360°\) for which
  1. \(\cos(\theta - 10°) = \cos 15°\), [3]
  2. \(\tan 2\theta = 0.4\), [5]
  3. \(2 \sin \theta \tan \theta = 3\). [6]
Edexcel C2 Q3
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 Q4
9 marks Moderate -0.3
  1. Write down formulae for sin (A + B) and sin (A - B). Using X = A + B and Y = A - B, prove that $$\sin X + \sin Y = 2 \sin \frac{X + Y}{2} \cos \frac{X - Y}{2}.$$ [4 marks]
  2. Hence, or otherwise, solve, for 0 ≤ θ < 360, $$\sin 40° + \sin 20° = 0.$$ [5 marks]
Edexcel C2 Q3
6 marks Moderate -0.8
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 Q7
10 marks Moderate -0.3
$$f(x) = 5\sin 3x°, \quad 0 \leq x \leq 180.$$
  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 Q8
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 Q5
8 marks Standard +0.3
Find, in degrees, the value of \(\theta\) in the interval \(0 \leq \theta < 360°\) for which $$2\cos^2 \theta - \cos \theta - 1 = \sin^2 \theta$$ Give your answers to 1 decimal place where appropriate. [8]
Edexcel C2 Q3
8 marks Moderate -0.8
Find all values of \(\theta\) in the interval \(0 \leq \theta < 360\) for which
  1. \(\cos(\theta + 75)^\circ = 0\). [3]
  2. \(\sin 2\theta^\circ = 0.7\), giving your answers to one decimal place. [5]
Edexcel C2 Q2
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 Q9
13 marks Moderate -0.3
  1. Solve, for \(0° < x < 180°\), the equation \(\sin (2x + 50°) = 0.6\), giving your answers to 1 d. p. [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 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 Q6
8 marks Standard +0.3
Find, in degrees, the value of \(\theta\) in the interval \(0 \leq \theta < 360°\) for which $$2\cos^2\theta - \cos\theta - 1 = \sin^2\theta.$$ Give your answers to \(1\) decimal place where appropriate. [8]
Edexcel C2 Q3
9 marks Moderate -0.8
The curve \(C\) has equation \(y = \cos \left( x + \frac{\pi}{4} \right)\), \(0 \leq x \leq 2\pi\).
  1. Sketch \(C\). [2]
  2. Write down the exact coordinates of the points at which \(C\) meets the coordinate axes. [3]
  3. Solve, for \(x\) in the interval \(0 \leq x \leq 2\pi\), \(\cos \left( x + \frac{\pi}{4} \right) = 0.5\), giving your answers in terms of \(\pi\). [4]
Edexcel C2 Q5
10 marks Standard +0.3
  1. Given that \(3 \sin x = 8 \cos x\), find the value of \(\tan x\). [1]
  2. Find, to 1 decimal place, all the solutions of \(3 \sin x - 8 \cos x = 0\) in the interval \(0 \leq x < 360°\). [3]
  3. Find, to 1 decimal place, all the solutions of \(3 \sin^2 y - 8 \cos y = 0\) in the interval \(0 \leq y < 360°\). [6]
OCR C2 Q9
12 marks Standard +0.2
    1. Write down the exact values of \(\cos \frac{1}{6}\pi\) and \(\tan \frac{1}{6}\pi\) (where the angles are in radians). Hence verify that \(x = \frac{1}{6}\pi\) is a solution of the equation $$2 \cos x = \tan 2x.$$ [3]
    2. Sketch, on a single diagram, the graphs of \(y = 2 \cos x\) and \(y = \tan 2x\), for \(x\) (radians) such that \(0 \leqslant x \leqslant \pi\). Hence state, in terms of \(\pi\), the other values of \(x\) between 0 and \(\pi\) satisfying the equation $$2 \cos x = \tan 2x.$$ [4]
    1. Use the trapezium rule, with 3 strips, to find an approximate value for the area of the region bounded by the curve \(y = \tan x\), the \(x\)-axis, and the lines \(x = 0.1\) and \(x = 0.4\). (Values of \(x\) are in radians.) [4]
    2. State with a reason whether this approximation is an underestimate or an overestimate. [1]
OCR C2 2007 January Q7
8 marks Moderate -0.8
    1. Sketch the graph of \(y = 2 \cos x\) for values of \(x\) such that \(0° \leq x \leq 360°\), indicating the coordinates of any points where the curve meets the axes. [2]
    2. Solve the equation \(2 \cos x = 0.8\), giving all values of \(x\) between \(0°\) and \(360°\). [3]
  2. Solve the equation \(2 \cos x = \sin x\), giving all values of \(x\) between \(-180°\) and \(180°\). [3]
OCR C2 Specimen Q5
8 marks Moderate -0.3
  1. Show that the equation \(15\cos^2\theta = 13 + \sin\theta\) may be written as a quadratic equation in \(\sin\theta\). [2]
  2. Hence solve the equation, giving all values of \(\theta\) such that \(0 \leq \theta \leq 360\). [6]
OCR MEI C2 2013 January Q9
5 marks Moderate -0.3
  1. Show that the equation \(\frac{\tan \theta}{\cos \theta} = 1\) may be rewritten as \(\sin \theta = 1 - \sin^2 \theta\). [2]
  2. Hence solve the equation \(\frac{\tan \theta}{\cos \theta} = 1\) for \(0° \leq \theta \leq 360°\). [3]
OCR MEI C2 2006 June Q7
5 marks Easy -1.3
  1. Sketch the graph of \(y = \cos x\) for \(0° \leq x \leq 360°\). On the same axes, sketch the graph of \(y = \cos 2x\) for \(0° \leq x \leq 360°\). Label each graph clearly. [3]
  2. Solve the equation \(\cos 2x = 0.5\) for \(0° \leq x \leq 360°\). [2]
OCR MEI C2 2008 June Q10
5 marks Standard +0.3
Showing your method, solve the equation \(2\sin^2\theta = \cos\theta + 2\) for values of \(\theta\) between \(0°\) and \(360°\). [5]
OCR MEI C2 2010 June Q8
5 marks Moderate -0.3
Showing your method clearly, solve the equation \(4 \sin^2 \theta = 3 + \cos^2 \theta\), for values of \(\theta\) between \(0°\) and \(360°\). [5]
OCR MEI C2 2013 June Q4
5 marks Moderate -0.8
  1. Starting with an equilateral triangle, prove that \(\cos 30° = \frac{\sqrt{3}}{2}\). [2]
  2. Solve the equation \(2 \sin \theta = -1\) for \(0 \leq \theta \leq 2\pi\), giving your answers in terms of \(\pi\). [3]
OCR MEI C2 2014 June Q9
3 marks Moderate -0.3
Solve the equation \(\tan 2\theta = 3\) for \(0° < \theta < 360°\). [3]
OCR MEI C2 2016 June Q7
5 marks Moderate -0.8
  1. Show that, when \(x\) is an acute angle, \(\tan x \sqrt{1 - \sin^2 x} = \sin x\). [2]
  2. Solve \(4 \sin^2 y = \sin y\) for \(0° \leq y \leq 360°\). [3]