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Quadratic Trigonometric Equations (DP IB Analysis & Approaches (AA)): Revision Note

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Quadratic Trigonometric Equations

How are quadratic trigonometric equations solved?

  • A quadratic trigonometric equation is one that includes either sin squared space theta, cos squared space theta or tan to the power of 2 space end exponent theta

  • Often the identityspace sin squared space theta plus cos squared space theta equals 1 can be used to rearrange the equation into a form that is possible to solve

    • If the equation involves both sine and cosine then the Pythagorean identity should be used to write the equation in terms of just one of these functions

  • Solve the quadratic equation using your GDC, the quadratic equation or factorisation

    • This can be made easier by changing the function to a single letter

      • Such as changing 2 cos squared space theta minus 3 cos space theta minus 1 equals 0 to 2 c squared minus 3 c minus 1 equals 0

  • A quadratic can give up to two solutions

    • You must consider both solutions to see whether a real value exists

    • Remember that solutions for sin θ = k and cos θ = k only exist for -1 ≤ k ≤ 1  

    • Solutions for tan θ = k exist for all values of k

  • Find all solutions within the given interval

    • There will often be more than two solutions for one quadratic equation

    • The best way to check the number of solutions is to sketch the graph of the function

Examiner Tips and Tricks

  • Sketch the trig graphs on your exam paper to refer back to as many times as you need to!

  • Be careful to make sure you have found all of the solutions in the given interval, being super-careful if you get a negative solution but have a positive interval

Worked Example

Solve the equation 11 sin space x blank – blank 7 blank equals 5 cos squared space x, finding all solutions in the range 0 space less or equal than space x space less or equal than space 2 straight pi.

aa-sl-3-6-5-quadratic-trig-equations-we-solution

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    Amber

    Author: Amber

    Expertise: Maths

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.