How To Solve Trigonometric Equations: 8 Steps (with Pictures)

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Terms of Use wikiHow is where trusted research and expert knowledge come together. Learn why people trust wikiHow How to Easily Solve Trigonometric Equations PDF download Download Article Explore this Article Steps Steps Other Sections Questions & Answers Video Related Articles References Reviewed by Joseph Meyer

Last Updated: November 16, 2024 References

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This article was reviewed by Joseph Meyer. Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. There are 9 references cited in this article, which can be found at the bottom of the page. This article has been viewed 267,714 times.

Did you get homework from your teacher that was about solving Trigonometric equations? Did you maybe not pay full attention in class during the lesson on Trigonometric questions? Do you even know what "Trigonometric" means? If you answered yes to these questions, then you don't need to worry because this wikiHow will teach you how to solve Trigonometric equations.

Steps

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  1. Step 1 Know the Solving concept. 1 Know the Solving concept.[1]
    • To solve a trig equation, transform it into one or many basic trig equations. Solving trig equations finally results in solving 4 types of basic trig equations.
  2. Step 2 Know how to solve basic trig equations. 2 Know how to solve basic trig equations.[2]
    • There are 4 types of basic trig equations:
    • sin x = a ; cos x = a
    • tan x = a ; cot x = a
    • Solving basic trig equations proceeds by studying the various positions of the arc x on the trig circle, and by using trig conversion table (or calculator). To fully know how to solve these basic trig equations, and similar, see book titled :"Trigonometry: Solving trig equations and inequalities" (Amazon E-book 2010).
    • Example 1. Solve sin x = 0.866. The conversion table (or calculator) gives the answer: x = Pi/3. The trig circle gives another arc (2Pi/3) that has the same sin value (0.866). The trig circle also gives an infinity of answers that are called extended answers.
    • x1 = Pi/3 + 2k.Pi, and x2 = 2Pi/3. (Answers within period (0, 2Pi))
    • x1 = Pi/3 + 2k Pi, and x2 = 2Pi/3 + 2k Pi. (Extended answers).
    • Example 2. Solve: cos x = -1/2. Calculators give x = 2 Pi/3. The trig circle gives another x = -2Pi/3.
    • x1 = 2Pi/3 + 2k.Pi, and x2 = - 2Pi/3. (Answers within period (0, 2Pi))
    • x1 = 2Pi/3 + 2k Pi, and x2 = -2Pi/3 + 2k.Pi. (Extended answers)
    • Example 3. Solve: tan (x - Pi/4) = 0.
    • x = Pi/4 ; (Answer)
    • x = Pi/4 + k Pi; ( Extended answer)
    • Example 4. Solve cot 2x = 1.732. Calculators and the trig circle give
    • x = Pi/12 ; (Answer)
    • x = Pi/12 + k Pi ; (Extended answers)
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  3. Step 3 Learn the Transformations used in solving trig equations. 3 Learn the Transformations used in solving trig equations.
    • To transform a given trig equation into basic trig ones, use common algebraic transformations (factoring, common factor, polynomial identities...), definitions and properties of trig functions, and trig identities. There are about 31, among them the last 14 trig identities, from 19 to 31, are called Transformation Identities, since they are used in the transformation of trig equations.[3] See book mentioned above.
    • Example 5: The trig equation: sin x + sin 2x + sin 3x = 0 can be transformed, using trig identities, into a product of basic trig equations: 4cos x*sin (3x/2)*cos (x/2) = 0. The basic trig equations to be solved are: cos x = 0 ; sin (3x/2) = 0 ; and cos (x/2) = 0.
  4. Step 4 Find the arcs whose trig functions are known. 4 Find the arcs whose trig functions are known.[4]
    • Before learning solving trig equations, you must know how to quickly find the arcs whose trig functions are known. Conversion values of arcs (or angles) are given by trig tables or calculators.[5]
    • Example: After solving, get cos x = 0.732. Calculators give the solution arc x = 42.95 degree. The trig unit circle will give other solution arcs that have the same cos value.
  5. Step 5 Graph the solution arcs on the trig unit circle. 5 Graph the solution arcs on the trig unit circle.[6]
    • You can graph to illustrate the solution arcs on the trig unit circle. The terminal points of these solution arcs constitute regular polygons on the trig circle. For examples:
    • The terminal points of the solution arcs x = Pi/3 + k.Pi/2 constitute a square on the trig unit circle.
    • The solution arcs x = Pi/4 + k.Pi/3 are represented by the vertexes of a regular hexagon on the trig unit circle.
  6. Step 6 Learn the Approaches to solve trig equations. 6 Learn the Approaches to solve trig equations.[7]
    • If the given trig equation contains only one trig function, solve it as a basic trig equation. If the given equation contains two or more trig functions there are 2 approaches in solving, depending on transformation possibility.
      • A. Approach 1.
    • Transform the given trig equation into a product in the form: f(x).g(x) = 0 or f(x).g(x).h(x) = 0, in which f(x), g(x) and h(x) are basic trig equations.
    • Example 6. Solve: 2cos x + sin 2x = 0. (0 < x < 2Pi)
    • Solution. Replace in the equation sin 2x by using the identity: sin 2x = 2*sin x*cos x.
    • cos x + 2*sin x*cos x = 2cos x*( sin x + 1) = 0. Next, solve the 2 basic trig functions: cos x = 0, and (sin x + 1) = 0.
    • Example 7. Solve: cos x + cos 2x + cos 3x = 0. (0 < x < 2Pi)
    • Solution: Transform it to a product, using trig identities: cos 2x(2cos x + 1 ) = 0. Next, solve the 2 basic trig equations: cos 2x = 0, and (2cos x + 1) = 0.
    • Example 8. Solve: sin x - sin 3x = cos 2x. (0 < x < 2Pi)
    • Solution: Transform it into a product, using trig identities: -cos 2x*(2sin x + 1) = 0. Then solve the 2 basic trig equations: cos 2x = 0, and (2sin x + 1) = 0.
      • B. Approach 2.
    • Transform the given trig equation into a trig equation having only one unique trig function as variable. There are a few tips on how to select the appropriate variable. The common variables to select are: sin x = t; cos x = t; cos 2x = t, tan x = t and tan (x/2) = t.
    • Example 9. Solve: 3sin^2 x - 2cos^2 x = 4sin x + 7 (0 < x < 2Pi).
    • Solution. Replace in the equation (cos^2 x) by (1 - sin^2 x), then simplify the equation:
    • 3sin^2 x - 2 + 2sin^2 x - 4sin x - 7 = 0. Call sin x = t. The equation becomes: 5t^2 - 4t - 9 = 0. This is a quadratic equation that has 2 real roots: t1 = -1 and t2 = 9/5. The second t2 is rejected since > 1. Next, solve: t = sin = -1 --> x = 3Pi/2.
    • Example 10. Solve: tan x + 2 tan^2 x = cot x + 2.
    • Solution. Call tan x = t. Transform the given equation into an equation with t as variable: (2t + 1)(t^2 - 1) = 0. Solve for t from this product, then solve the basic trig equation tan x = t for x.
  7. Step 7 Solve special types of trig equations. 7 Solve special types of trig equations.
    • There are a few special types of trig equations that require some specific transformations. Examples:
    • a*sin x+ b*cos x = c ; a(sin x + cos x) + b*cos x*sin x = c ;
    • a*sin^2 x + b*sin x*cos x + c*cos^2 x = 0
  8. Step 8 Learn the Periodic Property of trig functions. 8 Learn the Periodic Property of trig functions.[8]
    • All trig functions are periodic meaning they come back to the same value after a rotation for one period.[9] Examples:
      • The function f(x) = sin x has 2Pi as period.
      • The function f(x) = tan x has Pi as period.
      • The function f(x) = sin 2x has Pi as period.
      • The function f(x) = cos (x/2) has 4Pi as period.
    • If the period is specified in the problem/test, you have to only find the solution arc(s) x within this period.
    • NOTE: Solving trig equation is a tricky work that often leads to errors and mistakes. Therefore, answers should be carefully checked. After solving, you can check the answers by using a graphing calculator to directly graph the given trig equation R(x) = 0. The answers (real roots) will be given in decimals. For example, Pi is given by the value 3.14
    9. Advertisement

References

  1. ↑ https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-solve-for-a-side/a/unknown-side-in-right-triangle-w-trig
  2. ↑ https://www.purplemath.com/modules/solvtrig.htm
  3. ↑ https://www.khanacademy.org/math/precalculus/trig-equations-and-identities-precalc/using-trig-identities-precalc/v/examples-using-pythagorean-identities-to-simplify-trigonometric-expressions
  4. ↑ https://www.khanacademy.org/math/precalculus/trig-equations-and-identities-precalc/inverse-trig-functions-precalc/v/inverse-trig-functions-arcsin
  5. ↑ https://www.mathopenref.com/arcsin.html
  6. ↑ https://courses.lumenlearning.com/precalculus/chapter/unit-circle-sine-and-cosine-functions/
  7. ↑ https://mathbitsnotebook.com/Algebra2/TrigConcepts/TCEquationsMore.html
  8. ↑ https://www.analyzemath.com/trigonometry/properties.html
  9. ↑ https://www.youtube.com/watch?v=8Z60_yXX4xA

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About This Article

Joseph Meyer Reviewed by: Joseph Meyer Math Teacher This article was reviewed by Joseph Meyer. Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 267,714 times. 229 votes - 57% Co-authors: 18 Updated: November 16, 2024 Views: 267,714 Categories: Trigonometry In other languages Spanish Italian Russian Dutch
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