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Solving Equations Using Trigonometric Graphs (DP IB Analysis & Approaches (AA)): Revision Note

Amber

Written by: Amber

Reviewed by: Dan Finlay

Updated on

DP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SLDP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): HLDP Analysis & Approaches (AA): HLDP Analysis & Approaches (AA): SL

Using trigonometric graphs

How do I solve trig equations?

  • The inverse trig functions on your GDC will only give you one solution to a trig equation

    • This solution is called the principal value

      • e.g. sin to the power of negative 1 end exponent open parentheses 0.7 close parentheses is the principal value for the equation sin x equals 0.7

  • You can use the trig graphs and their symmetries to find the other solutions within an interval

How do I determine the number of solutions?

  • For sin x equals k and cos x equals k where negative 1 less than k less than 1 and k not equal to 0

    • There are always two solutions in any interval of length 360°

    • Divide the width of the interval by 360°

      • If it is a whole number then double it to get the number of solutions

      • Otherwise, double the closest whole numbers to find the minimum and maximum number of solutions

Examiner Tips and Tricks

Be careful when k equals negative 1 space or space 1, there is at least one solution every 360°. There could be 2 solutions depending on where the interval starts.

Be careful when k equals 0, there are at least two solutions every 360°. There could be 3 solutions depending on where the interval starts.

  • For tan x equals k

    • There is always one solution in any interval of length 180°

    • Divide the width of the interval by 180°

      • If it is a whole number then this is equal to the number of solutions

      • Otherwise, the closest whole numbers are the minimum and maximum number of solutions

How do I use trig graphs to solve trig equations?

  • STEP 1
    Sketch the graph for the given function and interval

    • Check whether you should be working in degrees or radians

    • Label the axes with the key values (0°, 90°, 180°, etc)

  • STEP 2
    Draw a horizontal line going through the y-axis at the relevant point

    • e.g. to solve sin x equals 0.7 draw the line y equals 0.7

  • STEP 3
    Find the principal value and mark it on the graph

  • STEP 4
    Use the symmetry and periodicity of the graph to find all the solutions in the interval

    • y equals sin x is symmetrical about x equals 90 degree and repeats every 360°

      • If x equals 50 degree a solution, then x equals 180 degree minus 50 degree is also a solution

    • y equals cos x is symmetrical about x equals 0 degree and repeats every 360°

      • If x equals 50 degree a solution, then x equals negative 50 degree is also a solution

    • y equals tan x repeats every 180°

      • If x equals 50 degree a solution, then x equals 50 degree plus 180 degree is also a solution

Worked Example

One solution to cos x = 0.5 is 60°. Find all the other solutions in the range -360° ≤ x ≤ 360°.

aa-sl-3-5-1-using-trig-graphs-we-solution-2

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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.