Multiple independent equations — all direct solve

Two or more completely separate trigonometric equations each solved directly (e.g. shifted sin/cos/tan equations, simple quadratics in trig), with no 'show that' or identity-proving component in any part.

14 questions · Moderate -0.4

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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 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 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]
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 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]
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]
Edexcel C3 Q1
8 marks Standard +0.3
  1. Find the exact value of \(x\) such that $$3 \arctan (x - 2) + \pi = 0.$$ [3]
  2. Solve, for \(-\pi < \theta < \pi\), the equation $$\cos 2\theta - \sin \theta - 1 = 0,$$ giving your answers in terms of \(\pi\). [5]
OCR PURE Q3
6 marks Moderate -0.8
  1. Solve the equation \(\sin^2\theta = 0.25\) for \(0° \leq \theta < 360°\). [3]
  2. In this question you must show detailed reasoning. Solve the equation \(\tan 3\phi = \sqrt{3}\) for \(0° \leq \phi < 90°\). [3]
WJEC Unit 3 2024 June Q2
11 marks Standard +0.3
  1. Find all values of \(\theta\) in the range \(0° < \theta < 360°\) satisfying $$3\cot\theta + 4\cosec^2\theta = 5.$$ [5]
  2. By writing \(24\cos x - 7\sin x\) in the form \(R\cos(x + \alpha)\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0° < \alpha < 90°\), solve the equation $$24\cos x - 7\sin x = 16$$ for values of \(x\) between \(0°\) and \(360°\). [6]
SPS SPS SM 2022 February Q3
8 marks Moderate -0.3
Solve each of the following equations, for \(0° \leqslant x \leqslant 180°\).
  1. \(2\sin^2 x = 1 + \cos x\). [4]
  2. \(\sin 2x = -\cos 2x\). [4]
SPS SPS FM 2023 October Q2
6 marks Moderate -0.8
Solve each of the following equations, for \(0° < x < 360°\).
  1. \(\sin \frac{1}{2}x = 0.8\) [3]
  2. \(\sin x = 3 \cos x\) [3]