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Solving Trig Equations (AQA GCSE Further Maths)

Revision Note

Paul

Written by: Paul

Reviewed by: Dan Finlay

Solving Basic Trig Equations

What are basic trig equations?

  • The types of equations dealt with here are those of the form

    • sin space x equals k

    • cos space x equals k

    • tan space x equals k

    • where k is a constant

  • (Most) scientific calculators will give you an answer to this type of equation

    • using inverse sin/cos/tan

      • e.g. x equals sin to the power of negative 1 end exponent space open parentheses k close parentheses

    • but trig equations usually have more than one solution

  • For inverse sin (sin-1) and inverse tan (tan-1)

    • (most) calculators will give an answer between -90° and 90°

  • For inverse cos (cos-1)

    • (most) calculators will give an answer between 0° and 180°

  • A question may require answers beyond these ranges

    • Typically, answers between 0° and 360° are required 

    • The properties (in particular, symmetry) of the trig graphs allow us to find all answers to trig equations

How are trigonometric equations of the form sin x = k solved?

  • The solutions to the equation sin x = 0.5 in the range 0° < x < 360° are x = 30° and x = 150°

    • If you like, check on a calculator that both sin 30° and sin 150° give 0.5

  • The first solution comes from your calculator (by taking inverse sin of both sides)

    • x = sin-1 (0.5) = 30°

  • The second solution comes from the symmetry of the graph y = sin x between 0° and 360°

    • Sketch the graph

    • Draw a vertical line from x = 30° to the curve, then horizontally across to another point on the curve, then vertically back to the x-axis again

    • By the symmetry of the curve, the new value of x is 180° - 30° = 150°

  • In general, if x° is an acute angle that solves sin x = k, then 180° - x° is the obtuse angle that solves the same equation

  • If/when a calculator gives x as a negative value when using inverse sin

    • extend your graph sketch for negative values of x (i.e. to the left of the y-axis)

cie-igcse-3-12-2-solving-trig-equations-1

How are trigonometric equations of the form cos x = k solved?

  • The solutions to the equation cos x = 0.5 in the range 0° < x < 360° are x = 60° and x = 300°

    • If you like, check on a calculator that both cos 60° and cos 300° give 0.5

  • The first solution comes from your calculator (by taking inverse cos of both sides)

    • x = cos-1 (0.5) = 60°

  • The second solution comes from the symmetry of the graph y = cos x between 0° and 360°

    • Sketch the graph

    • Draw a vertical line from x = 60° to the curve, then horizontally across to another point on the curve, then vertically back to the x-axis again

    • By the symmetry of the curve, the new value of x is 360° - 60° = 300°

  • In general, if x° is an angle that solves cos x = k, then 360° - x° is another angle that solves the same equation

  • (Most) calculators will not give x as a negative value when using inverse cos

cie-igcse-3-12-2-solving-trig-equations-2

How are trigonometric equations of the form tan x = k solved?

  • The solutions to the equation tan x = 1 in the range 0° < x < 360° are x = 45° and x = 225°

    • Check on a calculator that both tan 45° and tan 225° give 1

  • The first solution comes from your calculator (by taking inverse tan of both sides)

    • x = tan-1(1) = 45°

  • The second solution comes from the symmetry of the graph y = tan x between 0° and 360°

    • Sketch the graph

    • Draw a vertical line from x = 45° to the curve, then horizontally across to another point on the curve (a different “branch” of tan x), then vertically back to the x-axis again

    • The new value of x is 45° + 180° = 225° as the next “branch” of tan x is shifted 180° to the right

  • In general, if x° is an angle that solves tan x = k, then x° + 180° is another angle that solve the same equation

  • If/when a calculator gives x as a negative value when using inverse tan

    • extend your graph sketch for negative values of x (i.e. to the left of the y-axis)

cie-igcse-3-12-2-solving-trig-equations-3

Examiner Tips and Tricks

  • Use a calculator to check your solutions by substituting them into the original equation

    • e.g. 60° is a correct solution of cos x = 0.5 as cos 60° = 0.5 on a calculator
      but 330° is an incorrect solution as cos 330° ≠ 0.5

Worked Example

Solve sin x = 0.25 in the range 0° < x < 360°, giving your answers correct to 1 decimal place 

Use a calculator to find the first solution (by taking inverse sin of both sides)

x = sin-1(0.25) = 14.4775… = 14.48° to 2 dp

Sketch the graph of y = sin x and mark on (roughly) where x = 14.48 and y = 0.25 would be
Draw a vertical line up to the curve, then horizontally across to the next point on the curve, then vertically back down to the x-axis

cie-igcse-3-12-2-solving-trig-equations-4

Find this value using the symmetry of the curve (by taking 14.48 away from 180)

180° – 14.48° = 165.52°

Give both answers correct to 1 decimal place

x = 14.5° or x = 165.5°

Solving Quadratic Trig Equations

What is a quadratic trig equation?

  • A quadratic trig equation will have a 'squared' term involved in it

    • this will be either sin2x, cos2x or tan2x

    • there may be terms involving sin x, cos x or tan x as well

      • and possibly a constant term (i.e. a number)

  • Quadratic trig equations are very similar to normal quadratic equations

    • but the 'x' is replaced by 'sin x' (or 'cos x' or 'tan x')

  • Recall from the work on trig identities that sin2x means the same as (sin x)2

    • and likewise for cos/tan

How do I solve quadratic trig equations?

  • When first practising solving quadratic trig equations it may be helpful to replace sin x, cos x or tan x by a letter, y say

    • e.g. for the quadratic trig equation tan squared space x equals 8 tan space x minus 7 replacing "tan x" with "y" gives y squared equals 8 y minus 7

    • if you know what "hidden quadratics" are, this is the same process

  • Solve the quadratic equation as you normally would

    • the equation may need rearranging so it is in quadratic form

      • " a x squared plus b x plus c equals 0 "

      • e.g.  y squared minus 8 y plus 7 equals 0

    • solving the equation may involve factorising, completing the square or the quadratic formula

  • The answer(s) to the quadratic equation will not be the values for x (they are the values of y)

    • i.e.  they are the values of sin x, cos x or tan x

      • y = tan x in our example

  • Replace y with its trig function to form two basic trig equations to solve to find the values of x

    • e.g.  tan space x equals 7 and tan space x equals 1

    • Using inverse tan, the values for x in the range 0 degree less or equal than x degree less or equal than 180 degree are x equals 81.9 (1 d.p.) and x equals 45

    • Sketching the graph of y equals tan space x between 0° and 360° shows the other two solutions

      • x equals 81.9 plus 180 equals 261.9 degree and x equals 45 plus 180 equals 225 degree

  • In harder problems, one of the basic trig equations may have no possible solutions

    • e.g. cos space x equals 2

      • The graph y = cos x is always between -1 to 1 on the y-axis, so this trig equation has no solutions

Worked Example

Solve the equation 4 sin squared space x minus 1 equals 0 for 0 degree less or equal than x degree less or equal than 360 degree, giving your answers for x as exact values. 

Let y = sin x and rewrite the equation in y

4 y squared minus 1 equals 0 

This equation can be solved by rearranging to y squared equals 1 fourthbut you must remember plus-or-minus square root of blank end root)
A neater way to get two solutions is to use the difference of two squares

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

Solve the factorised equation (by setting each bracket equal to zero)

y equals 1 half or y equals negative 1 half 

Replace the y with sin x

sin space x equals 1 half or sin space x equals negative 1 half 

Write out the "first half" of this problem and solve it

sin space x space equals 1 half where 0 degree less than x less than 360 degree 

Use a calculator (or your knowledge of exact trig values) to find inverse sin

sin to the power of negative 1 end exponent open parentheses 1 half close parentheses equals 30 

Use a sketch of y equals sin space x to find the other solution (by symmetry

cie-igcse-3-12-2-solving-trig-equations-1

 Write out the two solutions so far

x = 30° or x = 150° 

Now write out the "second half" of this problem and solve it

sin space x equals negative 1 half where 0 degree less than x less than 360 degree 

From extending the sketch above in the negative x-axis, or using sin to the power of negative 1 end exponent open parentheses negative 1 half close parentheses, the first negative solution is

x = -30° 

But this is not in the range 0° to 360°
Adding 30° to 180° and subtracting 30° from 360° would give the correct values

x = 180° + 30° = 210° or x = 360° - 30° = 330° 

Combine these two solutions with the two found previously
It is good practice to write final solutions out in numerical order

x = 30, 150, 210, 330

Solving Trig Equations with Identities

How do I use trig identities to help solve trig equations?

  • In general, trig equations are easiest to solve when they involve one out of sin x, cos x or tan x only

  • For some equations involving different trig functions, you can often rewrite them back in terms of one only

    • To do this, use one (or both) of the identities

      • tan space theta identical to fraction numerator sin space theta over denominator cos space theta end fraction or sin squared space theta plus cos squared space theta identical to 1

  • The majority of questions will involve the second identity as this is frequently used with quadratic trig equations

    • It will be used in one of its rearranged forms

      • sin squared space theta equals 1 minus cos squared space theta or cos squared space theta equals 1 minus sin squared space theta

      • which will replace sin squared space theta terms (with cos2 ...) and cos squared space theta terms (with sin2 ...) respectively

      • choose whichever suits the question

      • e.g. The equation cos squared space x minus 3 sin space x plus 1 equals 0 can be rewritten as open parentheses 1 minus sin squared space x close parentheses minus 3 sin space x plus 1 equals 0
        which involves sin x terms only and so is easier to solve (after some rearranging)

  • The tan space theta identity is used less frequently and would normally appear in reverse

    • e.g. The equations fraction numerator sin space x over denominator cos space x end fraction equals 3 over 4 or  4 sin space x equals 3 space cos space x or 3 cos space x minus 4 space sin space x equals 0 etc can be rewritten as tan space x equals 3 over 4

    • this can be solved using a calculator and the graph of y equals tan space x

Examiner Tips and Tricks

  • Write out both identities for any trig equation questions in case they help

  • Sketch the appropriate graph

    • y = sin x, y = cos x or y = tan x

    • use the graph to ensure you find ALL solutions within the given interval

    • this is particularly important when a calculator gives an initial negative answer

Worked Example

Quadratic Trigonometric Equations Example Solution, A Level & AS Level Pure Maths Revision Notes

Worked Example

Solve the equation 3 cos space x equals 9 space sin space x minus 6 cos space x for 0 degree less or equal than x degree less or equal than 360 degree.

Add 6cos x to both sides to bring the cos x terms together 

9 space cos space x equals 9 space sin space x 

Divide both sides by 9

cos space x equals sin space x 

By considering the identity tan space x equals fraction numerator sin space x over denominator cos space x end fraction, divide both sides by cos x

fraction numerator cos space x over denominator cos space x end fraction equals fraction numerator sin space x over denominator cos space x end fraction 

Cancel the left-hand side (this does not cancel to give zero)

1 equals fraction numerator sin space x over denominator cos space x end fraction 

Use the identity tan space x equals fraction numerator sin space x over denominator cos space x end fraction to rewrite the right-hand side as tan x

1 equals tan space x 

The problem is now to solve the following

tan italic space x equals 1 where 0 degree less than x less than 360 degree 

Use tan to the power of negative 1 end exponent open parentheses 1 close parentheses, or your knowledge of exact trig values, to find the first x solution

x = 45 

Sketch the graph of y = tan x to find the other solution 

cie-igcse-3-12-2-solving-trig-equations-3

Write the two solutions in numerical order

x = 45 or x = 225

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    Paul

    Author: Paul

    Expertise: Maths Content Creator (Previous)

    Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths 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.