1.05o Trigonometric equations: solve in given intervals

1022 questions

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Edexcel C2 Q4
8 marks Moderate -0.8
    1. Sketch the curve \(y = \sin (x - 30)°\) for \(x\) in the interval \(-180 \leq x \leq 180\).
    2. Write down the coordinates of the turning points of the curve in this interval. [4]
  2. Find all values of \(x\) in the interval \(-180 \leq x \leq 180\) for which $$\sin (x - 30)° = 0.35,$$ giving your answers to 1 decimal place. [4]
Edexcel C2 Q4
8 marks Standard +0.8
Find all values of \(x\) in the interval \(0 \leq x < 360°\) for which $$2\sin^2 x - 2\cos x - \cos^2 x = 1.$$ [8]
Edexcel C2 Q8
10 marks Standard +0.3
  1. Given that \(\sin \theta = 2 - \sqrt{2}\), find the value of \(\cos^2 \theta\) in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are integers. [3]
  2. Find, in terms of \(\pi\), all values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\cos(2x - \frac{\pi}{6}) = \frac{1}{2}.$$ [7]
Edexcel C2 Q7
11 marks Moderate -0.3
$$f(x) = 2x^3 - 5x^2 + x + 2.$$
  1. Show that \((x - 2)\) is a factor of \(f(x)\). [2]
  2. Fully factorise \(f(x)\). [4]
  3. Solve the equation \(f(x) = 0\). [1]
  4. Find the values of \(\theta\) in the interval \(0 \leq \theta \leq 2\pi\) for which $$2\sin^3 \theta - 5\sin^2 \theta + \sin \theta + 2 = 0,$$ giving your answers in terms of \(\pi\). [4]
OCR C2 Q6
8 marks Moderate -0.8
$$f(x) = \cos 2x, \quad 0 \leq x \leq \pi.$$
  1. Sketch the curve \(y = f(x)\). [2]
  2. Write down the coordinates of any points where the curve \(y = f(x)\) meets the coordinate axes. [3]
  3. Solve the equation \(f(x) = 0.5\), giving your answers in terms of \(\pi\). [3]
OCR C2 Q3
6 marks Moderate -0.8
  1. Given that $$5 \cos \theta - 2 \sin \theta = 0,$$ show that \(\tan \theta = 2.5\) [2]
  2. Solve, for \(0 \leq x \leq 180\), the equation $$5 \cos 2x° - 2 \sin 2x° = 0,$$ giving your answers to 1 decimal place. [4]
OCR C2 Q5
8 marks Standard +0.3
  1. Given that \(\sin \theta = 2 - \sqrt{2}\), find the value of \(\cos^2 \theta\) in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are integers. [3]
  2. Find, in terms of \(\pi\), all values of \(x\) in the interval \(0 \leq x < \pi\) for which $$\cos 3x = \frac{\sqrt{3}}{2}.$$ [5]
OCR MEI C2 Q1
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 \(0 \leqslant \theta \leqslant 2\pi\), giving your answers in terms of \(\pi\). [3]
OCR MEI C2 Q5
5 marks Easy -1.2
  1. Sketch the graph of \(y = \cos x\) for \(0° \leqslant x \leqslant 360°\). On the same axes, sketch the graph of \(y = \cos 2x\) for \(0° \leqslant x \leqslant 360°\). Label each graph clearly. [3]
  2. Solve the equation \(\cos 2x = 0.5\) for \(0° \leqslant x \leqslant 360°\). [2]
OCR MEI C2 Q6
5 marks Moderate -0.8
  1. Sketch the graph of \(y = \sin \theta\) for \(0 \leqslant \theta \leqslant 2\pi\). [2]
  2. Solve the equation \(2 \sin \theta = -1\) for \(0 \leqslant \theta \leqslant 2\pi\). Give your answers in the form \(k\pi\). [3]
OCR MEI C2 Q7
4 marks Moderate -0.8
Sketch the curve \(y = \sin x\) for \(0° \leqslant x \leqslant 360°\). Solve the equation \(\sin x = -0.68\) for \(0° \leqslant x \leqslant 360°\). [4]
OCR MEI C2 Q8
5 marks Moderate -0.8
  1. Sketch the graph of \(y = \tan x\) for \(0° \leqslant x \leqslant 360°\). [2]
  2. Solve the equation \(4 \sin x = 3 \cos x\) for \(0° \leqslant x \leqslant 360°\). [3]
OCR MEI C2 Q9
4 marks Moderate -0.8
Sketch the graph of \(y = \sin x\) for \(0° \leqslant x \leqslant 360°\). Solve the equation \(\sin x = -0.2\) for \(0° \leqslant x \leqslant 360°\). [4]
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
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]
  3. Solve $$8 \cot \theta - 3 \tan \theta = 2,$$ for \(0 < \theta < 90°\), giving your answer to 1 decimal place. [5]
Edexcel C3 Q8
15 marks Standard +0.3
  1. Express \(2\cos\theta + 5\sin\theta\) in the form \(R\cos(\theta - \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\). Give the values of \(R\) and \(\alpha\) to 3 significant figures. [3]
  2. Find the maximum and minimum values of \(2\cos\theta + 5\sin\theta\) and the smallest possible value of \(\theta\) for which the maximum occurs. [2]
The temperature \(T\) °C, of an unheated building is modelled using the equation $$T = 15 + 2\cos\left(\frac{\pi t}{12}\right) + 5\sin\left(\frac{\pi t}{12}\right), \quad 0 \leq t < 24,$$ where \(t\) hours is the number of hours after 1200.
  1. Calculate the maximum temperature predicted by this model and the value of \(t\) when this maximum occurs. [4]
  2. Calculate, to the nearest half hour, the times when the temperature is predicted to be 12 °C. [6]
Edexcel C3 Q6
9 marks Standard +0.3
  1. Prove that $$\frac{1 - \cos 2\theta}{\sin 2\theta} \equiv \tan \theta, \quad \theta \neq \frac{n\pi}{2}, n \in \mathbb{Z}.$$ [3]
  2. Solve, giving exact answers in terms of \(\pi\), $$2(1 - \cos 2\theta) = \tan \theta, \quad 0 < \theta < \pi.$$ [6]
OCR C3 Q5
8 marks Standard +0.3
  1. Express \(3 \sin \theta + 2 \cos \theta\) in the form \(R \sin(\theta + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Hence solve the equation \(3 \sin \theta + 2 \cos \theta = \frac{5}{2}\), giving all solutions for which \(0° < \theta < 360°\). [5]
OCR C3 Q7
9 marks Standard +0.3
  1. Write down the formula for \(\cos 2x\) in terms of \(\cos x\). [1]
  2. Prove the identity \(\frac{4 \cos 2x}{1 + \cos 2x} = 4 - 2 \sec^2 x\). [3]
  3. Solve, for \(0 < x < 2\pi\), the equation \(\frac{4 \cos 2x}{1 + \cos 2x} = 3 \tan x - 7\). [5]
OCR C3 Q2
5 marks Standard +0.3
Solve, for \(0° < \theta < 360°\), the equation \(\sec^2 \theta = 4 \tan \theta - 2\). [5]
OCR C3 Q9
13 marks Challenging +1.2
  1. By first writing \(\sin 3\theta\) as \(\sin(2\theta + \theta)\), show that $$\sin 3\theta = 3 \sin \theta - 4 \sin^3 \theta.$$ [4]
  2. Determine the greatest possible value of $$9 \sin(\frac{10}{3}\alpha) - 12 \sin^3(\frac{10}{3}\alpha),$$ and find the smallest positive value of \(\alpha\) (in degrees) for which that greatest value occurs. [3]
  3. Solve, for \(0° < \beta < 90°\), the equation \(3 \sin 6\beta \cos 2\beta = 4\). [6]
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 Q8
11 marks Standard +0.3
  1. Express \(5 \cos x + 12 \sin x\) in the form \(R \cos(x - \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Hence give details of a pair of transformations which transforms the curve \(y = \cos x\) to the curve \(y = 5 \cos x + 12 \sin x\). [3]
  3. Solve, for \(0° < x < 360°\), the equation \(5 \cos x + 12 \sin x = 2\), giving your answers correct to the nearest \(0.1°\). [5]
OCR C3 Q5
8 marks Standard +0.3
  1. Express \(4 \cos \theta - \sin \theta\) in the form \(R \cos(\theta + \alpha)\), where \(R > 0\) and \(0° < \alpha < 90°\). [3]
  2. Hence solve the equation \(4 \cos \theta - \sin \theta = 2\), giving all solutions for which \(-180° < \theta < 180°\). [5]
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]