Linear Trigonometric Equations (CIE A Level Maths: Pure 1)

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Linear Trigonometric Equations

Solving linear trigonometric equations

  • You should have already seen two ways to solve linear trig equations
    • By sketching the graph (see Graphs of Trigonometric Functions) you can read off all the solutions in a given range (or interval)
    • By using trigonometric identities you can simplify harder equations

  • Another way to find solutions is by using the CAST diagram which shows where each function has positive solutions
  • You may be asked to use degrees or radians to solve trigonometric equations
    • Make sure your calculator is in the correct mode
    • Remember common angles
      • 90° is ½π radians
      • 180° is π radians
      • 270° is 3π/2 radians
      • 360° is 2π radians 


Linear Trigonometric Equations Diagram 1, A Level & AS Maths: Pure revision notes

How do I use the CAST diagram?

Linear Trigonometric Equations Diagram 2, A Level & AS Maths: Pure revision notes

What about more complicated trig equations? 

  • Some trig equations could involve a function of x or θ (see Transformations of Trigonometric Functions)
  • Functions could be in two forms, either y =  sin(θ ± k) or y = sin kθ
  • An easy way to solve an equation involving sin, cos or tan of (θ± k) or kθ is by transforming the range of the question

 Linear Trigonometric Equations Diagram 3, A Level & AS Level Pure Maths Revision Notes

Examiner Tip

  • Your calculator will only give you the principal value and you need to find all other solutions for the given interval.
  • Also, remember the CAST diagram only gives you some solutions, so again you may need to find more depending on the given interval.
  • It is entirely up to you how you solve a trig equation, but some ways are more helpful than others depending on the type of equation you are trying to solve

Worked example

Linear Trigonometric Equations Example Solution, A Level & AS Maths: Pure revision notes

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Lucy

Author: Lucy

Expertise: Head of STEM

Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels. Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all. Lucy has created revision content for a variety of domestic and international Exam Boards including Edexcel, AQA, OCR, CIE and IB.