How to Solve Trigonometric Equations: 8 Steps

2024 Author: Samantha Chapman | [email protected]. Last modified: 2023-12-17 09:13

A trigonometric equation is an equation that contains one or more trigonometric functions of the variable x. Solving for x means finding the values of x that, inserted in the trigonometric function, satisfy it.

• The solutions or values of arc functions are expressed in degrees or radians. For example: x = π / 3; x = 5π / 6; x = 3π2; x = 45 deg.; x = 37, 12 deg.; x = 178, 37 deg.
• Note: On the unit trig circle, the trig functions of each arc are the same trig functions of the corresponding angle. The trigonometric circle defines all the trigonometric functions on the arc variable x. It is also used as proof, in solving simple trigonometric equations or inequalities.

• Examples of trigonometric equations:

  • sin x + sin 2x = 1/2; tan x + cot x = 1,732
  • cos 3x + sin 2x = cos x; 2sin 2x + cos x = 1

  1. The unitary trigonometric circle.

     • It is a circle with radius = 1 unit, having O as its origin. The unit trigonometric circle defines 4 main trigonometric functions of the arc variable x which rotates counterclockwise on it.
     • When the arc, with value x, varies on the unit trigonometric circle:
     • The horizontal axis OAx defines the trigonometric function f (x) = cos x.
     • The vertical axis OBy defines the trigonometric function f (x) = sin x.
     • The vertical axis AT defines the trigonometric function f (x) = tan x.
     • The horizontal axis BU defines the trigonometric function f (x) = cot x.

  The unit trig circle is also used to solve basic trigonometric equations and inequalities by considering the various positions of the arc x on it

  Steps

  

  Step 1. Know the concept of resolution

  To solve a trig equation, turn it into one of the basic trig equations. Solving a trig equation ultimately consists of solving 4 types of basic trig equations

  

  Step 2. Figure out how to solve the basic equations

  • There are 4 types of basic trig equations:
  • sin x = a; cos x = a
  • tan x = a; cot x = a
  • Solving the basic trigonometric equations consists in studying the different positions of the arc x on the trigonometric circle, and using the conversion tables (or the calculator). To fully understand how to solve these basic equations, and the like, refer to the book: "Trigonometry: Solving trig equations and inequalities" (Amazon E-book 2010).
  • Example 1. Solve sin x = 0, 866. The conversion table (or calculator) returns the solution: x = π / 3. The trig circle has another arc (2π / 3) which has the same value for the sine (0, 866). The trigonometric circle provides an infinity of other solutions which are called extended solutions.
  • x1 = π / 3 + 2k. Pi, and x2 = 2π / 3. (Solutions with period (0, 2π))
  • x1 = π / 3 + 2k Pi, and x2 = 2π / 3 + 2k π. (Extended solutions).
  • Example 2. Solve: cos x = -1/2. The calculator returns x = 2 π / 3. The trigonometric circle gives another arc x = -2π / 3.
  • x1 = 2π / 3 + 2k. Pi, and x2 = - 2π / 3. (Solutions with period (0, 2π)
  • x1 = 2π / 3 + 2k Pi, and x2 = -2π / 3 + 2k.π. (Extended solutions)
  • Example 3. Solve: tan (x - π / 4) = 0.
  • x = π / 4; (Solutions with period π)
  • x = π / 4 + k Pi; (Extended solutions)
  • Example 4. Solve: cot 2x = 1,732. The calculator and the trigonometric circle returns:
  • x = π / 12; (Solutions with period π)
  • x = π / 12 + k π; (Extended solutions)

  

  Step 3. Learn the transformations to use to simplify trig equations

  • To transform a given trigonometric equation into a basic one, we use common algebraic transformations (factorization, common factors, polynomial identities, and so on), definitions and properties of trigonometric functions, and trigonometric identities. There are about 31, among which the last 14 trigonometric ones, from 19 to 31, are called Transformation Identities, since they are used to transform trigonometric equations. See the book indicated 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 trigonometric equations to be solved are: cos x = 0; sin (3x / 2) = 0; and cos (x / 2) = 0.

  

  Step 4. Find the arcs corresponding to the known trigonometric functions

  • Before learning how to solve trigonometric equations, you need to know how to quickly find the arcs of known trigonometric functions. The conversion values for arcs (or angles) are provided by trigonometric tables or by calculators.
  • Example: After solving, we get cos x = 0, 732. The calculator gives us the solution arc x = 42.95 degrees. The unit trigonometric circle will provide another solution: the arc which has the same value as the cosine.

  

  Step 5. Draw the arcs that are solution on the trigonometric circle

  • You can draw the arcs on the trig circle to illustrate the solution. The extreme points of these solution arcs constitute regular polygons on the trigonometric circle. Eg:
  • The extreme points of the arc solution x = π / 3 + k.π / 2 constitute a square on the trigonometric circle.
  • The solution arcs x = π / 4 + k.π / 3 are represented by the vertices of a regular hexagon on the unit trigonometric circle.

  

  Step 6. Learn the approaches to solving trigonometric equations

  • 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 trigonometric functions there are 2 ways to solve it, depending on the transformations available.

    A. Approach 1

  • Transform the given equation into a product of the form: f (x).g (x) = 0 or f (x).g (x).h (x) = 0, where f (x), g (x) and h (x) are basic trigonometric functions.
  • Example 6. Solve: 2cos x + sin 2x = 0 (0 <x <2π)
  • Solution. Replace sin 2x using the identity: sin 2x = 2 * sin x * cos x.
  • cos x + 2 * sin x * cos x = 2cos x * (sin x + 1) = 0. Then, solve the 2 basic trigonometric functions: cos x = 0, and (sin x + 1) = 0.
  • Example 7. Solve: cos x + cos 2x + cos 3x = 0. (0 <x <2π)
  • Solutions: Turn it into a product, using the trig identities: cos 2x (2cos x + 1) = 0. Then, solve the two basic trig equations: cos 2x = 0, and (2cos x + 1) = 0.
  • Example 8. Solve: sin x - sin 3x = cos 2x. (0 <x <2π)

  • Solution. Turn it into a product, using the 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 basic trig equation into a trig equation having a single trig function with variable. There are two 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 the equation (cos ^ 2 x) by (1 - sin ^ 2 x), then simplify the equation:
  • sin ^ 2 x - 2 - 2sin ^ 2 x - 4sin x - 7 = 0. Substitute sin x = t. The equation becomes: 5t ^ 2 - 4t - 9 = 0. It is a quadratic equation that has 2 real roots: t1 = -1 and t2 = 9/5. The second t2 is to be discarded as> 1. Then, solve: t = sin = -1 x = 3π / 2.
  • Example 10. Solve: tan x + 2 tan ^ 2 x = cot x + 2.
  • Solution. Substitute tan x = t. Transform the given equation into an equation with variable t: (2t + 1) (t ^ 2 - 1) = 0. Solve it for t from this product, then solve the basic trig equations tan x = t for x.

  

  Step 7. Solve particular types of trigonometric equations

  • There are some special types of trigonometric equations that require 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

  

  Step 8. Learn the periodic properties of trigonometric functions

  • All trigonometric functions are periodic, that is, they return to the same value after a rotation of a period. Examples:

    • The function f (x) = sin x has 2π as a period.
    • The function f (x) = tan x has π as a period.
    • The function f (x) = sin 2x has π as a period.
    • The function f (x) = cos (x / 2) has 4π as a period.
  • If the period is specified in the problem / test, you just have to find the solution arc (s) x within the period.
  • NOTE: Solving a trig equation is a difficult task that often leads to mistakes and mistakes. Hence, the answers must be checked carefully. After solving it, you can check the solutions by using a graph or a calculator to directly draw the trigonometric function R (x) = 0. The answers (real roots) will be given in decimals. For example, π is given by the value 3, 14.