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Linear Trigonometric Equations (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): SL

Trigonometric equations: sinx = k

How do I solve trigonometric equations?

  • Rearrange the equation to isolate the trig term

    • e.g. rearrange 5 sin x equals 3 to sin x equals 3 over 5

  • You can then find all the solutions using the trig graphs or the unit circle

  • The steps are summarised below

sinx = k & cosx = k

  • STEP 1
    Find the principal value using the inverse trig functions

    • You might have to use your GDC or your knowledge of exact trig values

      • e.g. for sin x equals 0.5 you find sin to the power of negative 1 end exponent open parentheses 0.5 close parentheses equals 30 degree

      • e.g. for cos x equals negative 0.5 you find cos to the power of negative 1 end exponent open parentheses negative 0.5 close parentheses equals 120 degree

  • STEP 2
    Find a second angle

    • For sin x equals k: subtract the principal angle from 180°

      • e.g. 180 degree minus 30 degree equals 150 degree

    • For cos x equals k: subtract the principal angle from 360°

      • e.g. 360 degree minus 120 degree equals 240 degree

  • STEP 3
    Find all the angles in the given interval by adding or subtracting multiples of 360° to your two angles

    • e.g. 30 degree plus 360 degree equals 390 degree and 150 degree plus 360 degree equals 510 degree

    • e.g. 120 degree minus 360 degree equals negative 240 degree and 240 degree minus 360 degree equals negative 120 degree

Examiner Tips and Tricks

For cos x equals k, you can also find a second angle by simply changing the sign of the principal value. For example, if x equals 120 degree is a solution to cos x equals negative 0.5, then so is x equals negative 120 degree.

tanx = k

  • STEP 1
    Find the principal value using the inverse trig functions

    • You might have to use your GDC or your knowledge of exact trig values

      • e.g. for tan x equals 1 you find tan to the power of negative 1 end exponent open parentheses 1 close parentheses equals 45 degree

  • STEP 2
    Find all the angles in the given interval by adding or subtracting multiples of 180° to your principal angle

    • e.g. 45 degree plus 180 degree equals 225 degree and 45 degree plus 360 degree equals 405 degree

Examiner Tips and Tricks

The equation could use degrees or radians. Make sure you change the angle mode of your GDC to match the question. And remember the two key conversions:

  • 180° = π radians

  • 360° = 2π radians

Worked Example

Solve the equation 2 cos space x space equals space minus 1 , finding all solutions in the range negative pi space less or equal than space x space less or equal than space 3 pi.

aa-sl-3-6-4-trig-equations-sinx--k-we-solution

Trigonometric equations: sin(ax + b) = k

How can I solve equations with transformations of trig functions?

  • STEP 1
    Make the substitution y equals a x plus b

    • e.g. for sin open parentheses 2 x plus 60 degree close parentheses equals fraction numerator square root of 3 over denominator 2 end fraction :

      • let y equals 2 x plus 60 degree

      • the equation becomes sin y equals fraction numerator square root of 3 over denominator 2 end fraction

  • STEP 2
    Write the interval in terms of the substitution

    • e.g. for the interval 0 degree less or equal than x less or equal than 360 degree:

      • multiply each part by 2 and add 60°

      • the interval becomes 0 degree less or equal than 2 x plus 60 degree less or equal than 780 degree

      • this is the same as 0 degree less or equal than y less or equal than 780 degree

  • STEP 3
    Find all the solutions for y that are in the interval

    • e.g. sin to the power of negative 1 end exponent open parentheses fraction numerator square root of 3 over denominator 2 end fraction close parentheses equals 60 degree

      • 180 degree minus 60 degree equals 120 degree

      • 60 degree plus 360 degree equals 420 degree

      • 120 degree plus 360 degree equals 480 degree

      • 60 degree plus 720 degree equals 780 degree

      • 120 degree plus 780 degree equals 900 degree ⨉ is outside the interval

  • STEP 4
    Find the value of x for each value of y using the substitution

    • y equals 2 x plus 60 degree rightwards double arrow x equals fraction numerator y minus 60 degree over denominator 2 end fraction

      • x equals 0 degree comma space 30 degree comma space 180 degree comma space 210 degree comma space 360 degree

Examiner Tips and Tricks

A common error that students make is dealing with the interval incorrectly. If the substation is y equals 2 x plus 60 degree, then you multiply the interval by 2 and then add 60°. However, a mistake is when students do the opposite: subtract 60° and then divide by 2.

Worked Example

Solve the equation 2 cos left parenthesis 2 x space minus space 30 degree right parenthesis space equals space minus 1, finding all solutions in the range negative 360 degree space less or equal than space x space less or equal than space 360 degree.

aa-sl-3-6-4-trig-equations-sinaxb--k-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.