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Solving Trigonometric Equations | MaThCliX, Schemes and Mind Maps of Trigonometry

Stanford UniversityTrigonometry

Solving Trigonometric Equations. Solving trigonometric equations involves many of the same skills as solving equations in general.

Typology: Schemes and Mind Maps

2022/2023

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Chapter5  IdentitiesandEquations
SolvingTrigonometricEquations


Solvingtrigonometricequationsinvolvesmanyofthesameskillsassolvingequationsingeneral.
Somespecificthingstowatchforinsolvingtrigonometricequationsarethefollowing:
 Arrangement.Itisoftenagoodideatogetarrangetheequationsothatalltermsareonone
sideoftheequalsign,andzeroisontheother.Forexample,tansintancanbe
rearrangedtobecometansintan0.
 Quadratics.Lookforquadraticequations.AnytimeanequationcontainsasingleTrig
functionwithmultipleexponents,theremaybeawaytofactoritlikeaquadraticequation.
Forexample,cos2cos1󰇛cos1󰇜.
 Factoring.Lookforwaystofactortheequationandsolvetheindividualtermsseparately.For
example,tansintantan󰇛sin1󰇜.
 TermswithNoSolution.Afterfactoring,sometermswillhavenosolutionandcanbe
discarded.Forexample,sin20requiressin2,whichhasnosolutionsincethe
sinefunctionnevertakesonavalueof2.
 Replacement.HavingtermswithdifferentTrigfunctionsinthesameequationisnota
problemifyouareabletofactortheequationsothatthedifferentTrigfunctionsarein
differentfactors.Whenthisisnotpossible,lookforwaystoreplaceoneormoreTrig
functionswithothersthatarealsointheequation.ThePythagoreanIdentitiesare
particularlyusefulforthispurpose.Forexample,intheequationcossin10,
coscanbereplacedby1sin,resultinginanequationcontainingonlyoneTrig
function.
 ExtraneousSolutions.Checkeachsolutiontomakesureitworksintheoriginalequation.A
solutionofonefactorofanequationmayfailasasolutionoverallbecausetheoriginal
functiondoesnotexistatthatvalue.SeeExample5.6below.
 InfiniteNumberofSolutions.Trigonometricequationsoftenhaveaninfinitenumberof
solutionsbecauseoftheirperiodicnature.Insuchcases,weappend“2”oranotherterm
tothesolutionstoindicatethis.SeeExample5.9below.
 SolutionsinanInterval.Becarefulwhensolutionsaresoughtinaspecificinterval.Forthe
interval󰇟0,2󰇜,therearetypicallytwosolutionsforeachfactorcontainingaTrigfunctionas
longasthevariableinthefunctionhasleadcoefficientof1(e.g.,orΞ).Ifthelead
coefficientisotherthan1(e.g.,5or5Ξ),thenumberofsolutionswilltypicallybetwo
multipliedbytheleadcoefficient(e.g.,10solutionsintheinterval󰇟0,2󰇜foraterminvolving
5î””).SeeExample5.5below,whichhas8solutionsontheinterval󰇟0,2󰇜.
Anumberofthesetechniquesareillustratedintheexamplesthatfollow.
Version 2.0
Page 47 of 109
January 1, 2016
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Solving Trigonometric Equations

Solving trigonometric equations involves many of the same skills as solving equations in general.

Some specific things to watch for in solving trigonometric equations are the following:

x Arrangement. It is often a good idea to get arrange the equation so that all terms are on one

side of the equal sign, and zero is on the other. For example, tan

àŹ¶

ʔ sin Ę”à”Œ tan

àŹ¶

ʔ can be

rearranged to become tan

àŹ¶

ʔ sin Ę”à”† tan

àŹ¶

x Quadratics. Look for quadratic equations. Any time an equation contains a single Trig

function with multiple exponents, there may be a way to factor it like a quadratic equation.

For example, cos

àŹ¶

Ę”à”… 2 cos Ę”à”… 1 à”Œ

cos Ę”à”… 1

àŹ¶

x Factoring. Look for ways to factor the equation and solve the individual terms separately. For

example, tan

àŹ¶

ʔ sin Ę”à”† tan

àŹ¶

Ę”à”Œ tan

àŹ¶

ʔ áˆșsin Ę”à”† 1ሻ.

x Terms with No Solution. After factoring, some terms will have no solution and can be

discarded. For example, sin Ę”à”† 2 à”Œ 0 requires sin Ę”à”Œ 2, which has no solution since the

sine function never takes on a value of 2.

x Replacement. Having terms with different Trig functions in the same equation is not a

problem if you are able to factor the equation so that the different Trig functions are in

different factors. When this is not possible, look for ways to replace one or more Trig

functions with others that are also in the equation. The Pythagorean Identities are

particularly useful for this purpose. For example, in the equation cos

àŹ¶

Ę”à”† sin Ę”à”† 1 à”Œ 0,

cos

àŹ¶

ʔ can be replaced by 1 à”† sin

àŹ¶

ʔ, resulting in an equation containing only one Trig

function.

x Extraneous Solutions. Check each solution to make sure it works in the original equation. A

solution of one factor of an equation may fail as a solution overall because the original

function does not exist at that value. See Example 5.6 below.

x Infinite Number of Solutions. Trigonometric equations often have an infinite number of

solutions because of their periodic nature. In such cases, we append â€œßšĘŠà”…2 ” or another term

to the solutions to indicate this. See Example 5.9 below.

x Solutions in an Interval. Be careful when solutions are sought in a specific interval. For the

interval ሟ0, 2ߚሻ, there are typically two solutions for each factor containing a Trig function as

long as the variable in the function has lead coefficient of 1 (e.g., ʔ or ξ). If the lead

coefficient is other than 1 (e.g., ʔ5 or 5ξ), the number of solutions will typically be two

multiplied by the lead coefficient (e.g., 10 solutions in the interval ሟ0, 2ߚሻ for a term involving

ʔ5). See Example 5.5 below, which has 8 solutions on the interval ሟ0, 2ߚሻ.

A number of these techniques are illustrated in the examples that follow.

Solving Trigonometric Equations – Examples

Example 5.4: Solve for ʔ on the interval ሟ0, 2ߚሻ: cos

àŹ¶

Ę”à”… 2 cos Ę”à”… 1 à”Œ 0

The trick on this problem is to recognize the expression as a quadratic equation. Replace the

trigonometric function, in this case, cos ʔ, with a variable, like ʑ, that will make it easier to see

how to factor the expression. If you can see how to factor the expression without the trick, by all

means proceed without it.

Let Ę‘à”Œ cos ʔ, and our equation becomes: ʑ

àŹ¶

This equation factors to get:

àŹ¶

Substituting cos ʔ back in for ʑ gives:

cos Ę”à”… 1

àŹ¶

And finally: cos Ę”à”… 1 à”Œ 0 ⇒ cos Ę”à”Œ à”†

The only solution for this on the interval ሟ0, 2ߚሻ is: àąžà”ŒàŁŠ

Example 5.5: Solve for ʔ on the interval ሟ0, 2ߚሻ: sin 4ʔ à”Œ

√

àŹ·

àŹ¶

When working with a problem in the interval ሟ0, 2ߚሻ that involves a function of ʔʇ , it is useful to

expand the interval to ሟ0, 2ßšĘ‡áˆ» for the first steps of the solution.

In this problem, ʇ à”Œ 4 , so we want all solutions to sin ʑ à”Œ

âˆšàŹ·

àŹ¶

where ʔ4 à”Œ ʑ is an angle in the

interval ሟ0, 8ߚሻ. Note that, beyond the two solutions suggested by the diagram, additional

solutions are obtained by adding multiples of 2ßš to those two solutions.

Using the diagram at left, we get the following solutions:

Then, dividing by 4, we get:

And simplifying, we get:

Note that there are 8 solutions

because the usual number of

solutions (i.e., 2 ) is increased

by a factor of ʇ à”Œ 4.

Solving Trigonometric Equations – Examples

Example 5.9: Solve for all solutions of ʔ: 2 sin ʔ à”† √ 3 à”Œ 0

2 sin ʔ à”Œ √

sin ʔ à”Œ

Example 5.10: Solve for all solutions of ʔ: tan ʔ sec Ę”à”Œ à”†2 tan ʔ

tan ʔ sec Ę”à”… 2 tan Ę”à”Œ 0 áˆșsec Ę”à”… 2ሻ à”Œ 0

tan ʔ áˆșsec Ę”à”… 2ሻ à”Œ 0 sec Ę”à”Œ à”†

tan Ę”à”Œ 0 or áˆșsec Ę”à”… 2ሻ à”Œ 0 cos Ę”à”Œ à”†

àŹ”

àŹ¶

àŹ¶à°—

àŹ·

߹ʊ2 à”… or à”Œ ʔ

àŹžà°—

àŹ·

Collecting the various solutions, àąžâˆˆ áˆŒàŁŠàą” ሜ âˆȘ ቄ

à«›àŁŠ

૜

à«àŁŠ

૜

Note: the solution involving the tangent function has two answers in the interval ሟ0, 2ߚሻ.

However, they are ßš radians apart, as most solutions involving the tangent function are.

Therefore, we can simplify the answers by showing only one base answer and adding ʊ ߹ , instead

of showing two base answers that are ߹ apart, and adding ߹ʊ2 to each.

For example, the following two solutions for tan Ę”à”Œ 0 are telescoped into the single solution

given above:

The drawing at left illustrates the two

angles in ሟ0, 2ߚሻ for which sin ʔ à”Œ

âˆšàŹ·

àŹ¶

. To

get all solutions, we need to add all

integer multiples of ßš2 to these solutions.

So,

àŁŠ

૜

à«›àŁŠ

૜