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

Amber

Written by: Amber

Reviewed by: Dan Finlay

Updated on

DP Analysis & Approaches (AA): SLDP Analysis & Approaches (AA): HL

Quadratic trigonometric equations

How do I solve quadratic trigonometric equations involving a single ratio?

  • STEP 1
    Make a substitution equal to the trig ratio

    • e.g. for 2 cos squared x plus 3 cos x minus 2 equals 0

      • let y equals cos x

      • the equation becomes 2 y squared plus 3 y minus 2 equals 0

  • STEP 2
    Solve the new quadratic equation

    • You can factorise, complete the square, use the formula or use your GDC

      • open parentheses 2 y minus 1 close parentheses open parentheses y plus 2 close parentheses equals 0

      • y equals 1 half or y equals negative 2

  • STEP 3
    Replace the variable with the substitution and reject any that do not have solutions

    • Remember the values of sin and cos range from -1 to 1

      • cos x equals 1 half

      • cos x equals negative 2 has no solutions as negative 2 less than negative 1

  • STEP 4
    Solve the trig equation(s) for the given interval

How do I solve trigonometric equations involving multiple ratios?

If your equation involves tan and another ratio

  • Use tan x equals fraction numerator sin x over denominator cos x end fraction to write the equation in terms of sin and cos only

    • e.g. 2 sin x equals tan x becomes 2 sin x equals fraction numerator sin x over denominator cos x end fraction

  • Get rid of any fractions

    • e.g. 2 sin x cos x equals sin x

  • Make the equation equal to zero

    • 2 sin x cos x minus sin x equals 0

  • Factorise the expression

    • e.g. sin x open parentheses 2 cos x minus 1 close parentheses equals 0

  • Set each factor equal to zero and solve

    • e.g. sin x equals 0 and cos x equals 1 half

If your equation involves sin2 and/or cos2

  • Use sin squared x plus cos squared x equals 1 to write the equation in terms of only one ratio

    • e.g. 3 cos squared x plus 4 sin x equals 4 can be written as 3 open parentheses 1 minus sin squared x close parentheses plus 4 sin x equals 4

  • Make the equation equal to zero

    • e.g. 3 minus 3 sin squared x plus 4 sin x equals 4 becomes 3 sin squared x minus 4 sin x plus 1 equals 0

  • Solve the quadratic trig equation

    • e.g. sin x equals 1 third and sin x equals 1

Examiner Tips and Tricks

If there is a linear trig term and a quadratic trig term, then change the quadratic one to match the linear one. For example, if you have sin squared x and cos x, then use sin squared x equals 1 minus cos squared x.

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 Content Creator

    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.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.