Lesson Solving trigonometric equations
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DEFINITION. A trig equation is an equation containing one or many trig functions of the variable arc x that rotates on the trig unit circle. Solving for x means finding the values of the variable trig arc x whose trig functions make the trig equation true. Examples of trig equation: sin ( x + 30 degree) = 0.75 ; tan (x + pi/3) = 1.5 ; sin 2x + cos x = 1 ; tan x + cot x = 1.732 sin x + sin 2x = sin 3x ; cos x + cos 2x + cos 3x = 0 ; sin x - sin 3x = cos 2x Answers, or values of the solution arcs, are expressed in degrees or radians. Examples: x = 45 degree ; x = 47.24 degree ; x = 25.59 degree x = Pi/5 ; x = 3Pi /4 ; x = 7Pi/12 THE TRIG UNIT CIRCLE. It is a circle with radius R = 1 unit with the origin O. The unit circle defines the trig functions of the variable arc x that rotates counter-clockwise on the trig circle. When the arc AM, with value x (in radians or degrees), varies on the trig unit circle: The horizontal axis OAx defines the function f(x) = cos x. The vertical axis OBy defines the function f(x) = sin x. The vertical axis At defines the function f(x) = tan x. The horizontal axis Bu defines the function f(x) = cot x. THE PERIODIC PROPERTY OF TRIG FUNCTIONS All trig functions f(x) are periodic meaning they come back to the same value after the arc x rotates counterclockwise one period on the unit circle. Examples: The trig function f(x) = sin x has 2Pi as period. The trig function f(x) = tan x has Pi as period. The trig function f(x) = sin 2x has Pi as period. The trig function f(x) = cos x/2 has 4Pi as period. CONCEPT FOR SOLVING TRIG EQUATIONS To solve a trig equation transform it into one or many basic trig equations. Solving trig equations finally results in solving basic trig equations. BASIC TRIG EQUATIONS They are also called "Trig equations in simplest form". There are 4 types of common basic trig equations: sin x = a; cos x = a; tan x = a; cot x = a. Solving a basic trig equation proceeds by considering the various positions of the arc x on the trig unit circle and by using the trig conversion tables (or calculators). Example 1. Solve sin x = 0.866. The 2 answers are given by the trig unit circle and calculators: Answer 1: x1 = Pi/3 ; Extended answer: x1 = Pi/3 + 2k.Pi Answer 2: x2 = 2Pi/3 ; Extended answer: x2 = 2Pi/3 + 2k.Pi Example 2. Solve: cos x = -1/2. Two answers are given by the unit circle and conversion table: Answer 1: x1 = 2Pi/3 ; Extended answer: x1 = 2Pi/3 + 2k.Pi Answer 2: x2 = -2Pi/3 ; Extended answer: x2 = -2Pi/3 + 2k.Pi Example 3. Solve cos x = 0.732. Two answers given by calculator and the unit circle: Answer 1: x1 = 42.95 degree ; Extended answer: x1 = 42.95 degree + k.360 degree Answer 2: x2 = -42.95 degree ; Extended answer: x2 = -42.95 degree + k.360 degree Example 4. Solve: cot 2x = 1.732. Trig table and unit circle give: 2x = Pi/6 ; Extended: 2x = Pi/6 + K.Pi Answer: x = Pi/12 ; Extended answer: x = Pi/12 + k.Pi/2. Example 5. Solve: sin(x - 20 deg.) = 0.5. Trig table and trig unit circle gives: 1) sin (x - 20) = sin 30 deg.-------- 2) sin (x - 20) = sin (180 - 30) x - 20 = 30 deg. ; ------------------ x - 20 = 150 deg. x = 50 deg. ; ----------------------- x = 170 deg. Extended x = 50 deg. + k.360 deg. ; -- Extended x = 170 deg.+ k.360 deg. Example 6. Solve: sin 2x = cos 3x. The unit circle gives 2 answers: 1) sin 2x = sin (Pi/2 - 3x) ; ------ 2) sin 2x = sin (Pi - Pi/2 + 3x) 2x = Pi/2 - 3x ; -------------------- 2x = Pi/2 + 3x 5x = Pi/2 ; ------------------------- -x = Pi/2 x = Pi/10 ; ------------------------- x = -Pi/2 Extended x = Pi/10 + k.2Pi ; --------- Extended: x = -Pi/2 + k.2Pi. TRANSFORMATION USED IN SOLVING TRIG EQUATIONS. To transform a trig equation into basic trig equations, use common algebraic transformations (factoring, common factor, polynomial identities...), definition and properties of trig functions, and trig identities (the most needed). There are 14 common trig identities, called "transformation identities", that are used for the transformation of trig equations. See book titled:"Solving trigonometric equations and inequalities" (Amazon e-book 2010). Example. The trig equation sin x + sin 2x + sin 3x = 0 can be transformed, using trig identities, into a product of many basic trig equations: 4cos x.sin 3x/2.cos x/2 = 0. Example. The trig equation cos x + cos 2x + cos 3x = 0 can be transformed, using trig identities, into a product of basic trig equation: cos 2x(2cos x + 1) = 0. Example. Transform into a product (sin a + cos a). sin a + cos a = sin a + sin (Pi/2 - a) = 2sin (Pi/4).sin (a + Pi/4) Example. Transform (sin 2a - sin a) into a product: sin 2a - sin a = 2sin a.cos a - sin a = sin a.(2cos a - 1). GRAPHING THE SOLUTION ARCS ON THE UNIT CIRCLE We can graph to illustrate the solution arcs on the trig unit circle. The terminal points of the solution arcs constitute regular polygons on the trig unit circle. Example: The terminal points of the solution arcs x = Pi/3 + k.Pi/2 constitute a square on the unit circle. Example. The solution arcs x = Pi/4 + k.Pi/3 are represented by the vertexes of a regular hexagon on the unit circle. METHODS FOR SOLVING TRIG EQUATIONS. There are 2 methods depending on transformation possibilities. METHOD 1. Transform the given trig equation into a product of basic trig equations. Next, solves these basic trig equations to get all the solution arcs. Example 7. Solve sin 2x + 2cos x = 0. Solution: First, transform: sin 2x + 2cos x = 2sin x.cos x + 2cos x = 2cos x(sin x + 1) = 0. Next, solve the 2 basic trig equations: cos x = 0 and sin x = -1. Example 8. Solve: cos x + cos 2x + cos 3x = 0. Solution. First, use trig identities to transform: (cos x + cos 3x) + cos 2x = 2cos 2x.cos x + cos 2x = cos 2x(2cos x + 1) = 0. Next, solve the 2 basic trig equations: cos 2x = 0 and cos x = -1/2. Example 9. Solve sin x - sin 3x = cos 2x. Solution. Using trig identity transform the equation. (sin x - sin 3x) - cos 2x = 2cos 2x.sin x - cos 2x = cos 2x(2sin x - 1) = 0. Next, solve the 2 basic trig equations: cos 2x = 0 and sin x = 1/2. METHOD 2. If the given trig equation contains 2 or more trig functions, transform it into an equation containing only one trig function as variable. There are a few tips on how to select the trig function variable. The common variables to select are: sin x = t; cos x = t; cos 2x = t; tan x = t; tan x/2 = t. Example 10. Solve: 3sin^2 x + 2cos x - 2 = 0. Call cos x = t. Solution. Replace (sin^2 x) by (1 - cos^2 x) = 1 - t^2. The equation becomes: 3(1 - t^2) + 2t - 2 = -3t^2 + 2t + 1 = 0. This is a quadratic equation with 2 real roots: 1 and -1/3. Next, solve the 2 basic trig equations: t = cos x = 1 and t = cos x = -1/3. Example 11. Solve: tan x + 2tan^2 x - cot x - 2 = 0. Solution. Call tan x = t, the equation becomes: t + 2t^2 - 1/t - 2 = 0 = t^2 + 2t^3 - 1 - 2t = t^2(1 + 2t) - (1 + 2t) = (1 + 2t)(t^2 - 1) = 0. Next, solve the 2 basic trig equations: (t = tan x = -1/2) and (t^2 = tan^2 x = 1). Example 12. Solve cos 2x - 3sin x - 2 = 0. Solution. Call sin x = t and replace (cos 2x) by (1 - 2sin^2 x). (1 - 2t^2) - 3t - 2 = -2t^2 - 3t - 1 = 0. Quadratic equation with 2 real roots: -1 and -1/2. Next, solve the 2 basic trig equations: t = sin x = -1 and t = sin x = -1/2. SOLVING SPECIAL TYPES OF TRIG EQUATIONS. There are a few types of trig equations that require specific transformations. Examples: a.sin x + b.cos x = c a(sin x + cos x) + b.sin x.cos x = c a.sin^2 x + b.sin x.cos x + c.cos^2 x = 0 THE COMMON PERIOD OF A TRIG EQUATION Unless specified in home-works/tests, the trig equation f(x) = 0 must be solved, at least, within a common period. This means we must find all the solution arcs x within this common period. The common period is the least multiple of all the periods of the trig functions presented in the equation. Examples: The trig equation f(x) = cos x + 2tan x - 2 = 0 has 2Pi as common period. The equation f(x) = tan x + 2cot x = 0 has Pi as common period. The equation f(x) = cos2x + sin x = 0 has 2Pi as common period. The equation f(x) = sin 2x + cos x - sin x/2 = 0 has 4Pi as common period. CHECKING ANSWERS BY GRAPHING CALCULATORS AFTER SOLVING. Solving trig equations is a tricky work that often leads to errors and mistakes. After solving, you may check the answers by using graphing calculators. Using appropriate calculator setup, graph the function f(x). The roots of f(x) = 0 will be given in decimals. For examples, Pi is given as 3.14; 360 degree is given as 6.28. For more details, see the last chapter of the trig book mentioned above.
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