Quadratic in sin²/cos²/tan²

Solve equations that are quadratic in sin² θ, cos² θ, or tan² θ (or equivalent substitutions like x = sin² θ).

8 questions · Moderate -0.3

1.05o Trigonometric equations: solve in given intervals
Sort by: Default | Easiest first | Hardest first
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]
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 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 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 H240/02 2020 November Q4
5 marks Standard +0.3
In this question you must show detailed reasoning. Solve the equation \(3\sin^4 \phi + \sin^2 \phi = 4\), for \(0 \leq \phi < 2\pi\), where \(\phi\) is measured in radians. [5]
AQA AS Paper 1 2023 June Q4
5 marks Moderate -0.3
It is given that \(5\cos^2 \theta - 4\sin^2 \theta = 0\)
  1. Find the possible values of \(\tan \theta\), giving your answers in exact form. [3 marks]
  2. Hence, or otherwise, solve the equation $$5\cos^2 \theta - 4\sin^2 \theta = 0$$ giving all solutions of \(\theta\) to the nearest \(0.1°\) in the interval \(0° \leq \theta \leq 360°\) [2 marks]
AQA AS Paper 2 2018 June Q4
4 marks Moderate -0.3
Solve the equation \(\tan^2 2\theta - 3 = 0\) giving all the solutions for \(0° \leq \theta \leq 360°\) [4 marks]