0
Trigonometric equations often have infinitely many solutions, due to the periodicity of the functions. Take a simple example, sin x = 0. This equation is satisfied by an angle of zero, but also by an angle of pi, 2 x pi, 3 x pi, etc. (this is in radians; the equivalent in degrees would be 0°, 180°, 360°, etc.). Once you find two base solutions (in this case 0 and pi), repeatedly adding the length of the period (in this case, 2 pi, equivalent to 360°) will give you additional solutions.
Add your answer:
Earn +20 pts
Why is the period of a function important when solving a trig equation?
there can be more than 1 answer for some trig equations and you must use your knowledge of periodicity to get the answers.
Who are the important people in trigonometry?
The students and teacher of my trig. class
Is the avd algreba or algebra trig is harder?
trig
Is sine a trig function?
Yes, sine is a trig function, it is opposite over hypotenuse.
How is trigonometry relevant to our everyday life?
Surveying is all trig. Much trig in physics and engineering,
Why is the period of a function important when solving a trig equation?
there can be more than 1 answer for some trig equations and you must use your knowledge of periodicity to get the answers.
What are the ways on how to solve trigonometric equations?
Generalities.A trig equation contains one or many trig functions of the variable arc x. Solving for x means finding the values of the trig arcs x whose trig functions make the equations true.Example of trig equations:sin (x + Pi/3) = 0.75 ; sin 2x + cos x = 1 ; tan x + 2 cot x = 3 ; tan x + cot x = 1.732.sin x + sin 3x = 1. 5 ; sin x + sin 2x + sin 3x = cos x + cos 2x + cos 3x ;The answers, or values of the solution arcs x, are expressed in terms of radians or degrees:x = Pi/3 ; x = 137 deg. ; x = 2Pi/3 + 2k.Pi ; x = - 17. 23 deg. ; x = 360 deg.The Trig Unit CircleIt is a circle with radius R = 1 unity, and with an origin O. This unit circle defines all trig functions of the variable arc x that rotates counterclockwise on it.When the arc AM, with value x, rotates on the unit circle,The horizontal axis OAx defines the trig function f(x) = cos x.The vertical axis OBy defines the trig function f(x) = sin x.The vertical axis AT defines the trig function f(x) = tan x.The horizontal axis BU defines the trig function f(x) = cot xThe trig unit circle will be used as proofs for solving basic trig equations and trig inequalities.The periodic property of all trig functions.All trig functions are periodic meaning they come back to the same values when the arc x completes one period of rotation on the trig unit circle.Examples:The trig functions f(x) = sin x and f(x) = cos x have 2Pi as periodThe trig function f(x) = tan x and f(x) = cot x have Pi as period.Find the arcs whose trig functions are known.You must know how to find the values of the arcs when their trig functions are known. Conversion values are given by calculators or trig tables.Example: After solving, you get cos x = 0.732. Calculators (or trig table) gives x = 42.95 deg.. The Unit Circle will give an infinity of other arcs x that have the same cos value. These values are called extended answers.Example: Get sin x = 0.5. Trig table gives x = Pi/6. The unit circle give an infinity of extended answers.Concept for solving trig equations.To solve a trig equations, transform it into one or many basic trig equations.Basic trig equations.There are 4 of them. They are also called "trig equations in simplest form".sin x = a ; cos x = a (a is a given number)tan x = a ; cot x = aSolving basic trig equations.The solving method proceeds by considering the various positions of the variable arc x, rotating on the trig circle, and by using calculators (or trig tables).Example 1. Solve sin x = 0.866Solution. There are 2 answers given by calculators and the trig circle:x = Pi/3 ; x = 2Pi/3 (answers)x = Pi/3 + 2k.Pi ; x = 2Pi/3 + 2k.Pi (extended answers)Example 2. Solve cos x = 0.5Solution. 2 answers given by the trig table and the trig circle:x = 2Pi/3 ; x = - 2Pi/3 (answers)x = 2Pi/3 + 2k.Pi ; x = -2Pi/3 + 2k.Pi (extended answers)Note. The answer x = - 2Pi/3 can be replaced by x = 2Pi - 2Pi/3 = 4Pi/3.How to transform a given trig equation into basic trig equations.You may use:- common algebraic transformations, such as factoring, common factor, polynomials identities....- definitions and properties of trig functions...- trig identities (the most needed)Common Trig Identities.There are about 31 of them. Among them, the last 14 identities, from #19 to #31, are called "Transformation Identities" since they are necessary tools to transform a given trig equation into many basic ones. See book titled "Solving trig equations and inequalities" (Amazon e-book 2010)Examples of trig identities: sin^2 a + co^2 a = 1 ; sin 2a = 2sin a.cos a ;1 - cos 2a = 2 sin^2 a ; cos a = (1 - t^2)/(1 + t^2)Methods to solve trig equations.There are 2 common methods to solve a trig equation, depending on transformation possibilities.Method 1. Transform it into a product of many basic trig equations, by usingcommon transformation tools or by using trig identities.Example 3. Solve 2cos x + sin 2x = 0.Solution. Replace sin 2x by 2sin x.cos x (Trig Identity #10)2cos x + sin 2x = 2cos x + 2sin x.cos x = 2cos x(1 + sin x).Next, solve the 2 basic trig equations: cos x = 0 and sin x + 1 = 0.Example 4. Solve cos x + cos 2x + cos 3x = 0.Solution. Using trig identity #26, transform it into a product of 2 basic trig equations: cos 2x (2 cos x + 1) = 0. Next, solve the 2 basic trig equations: cos 2x = 0 and cos x = -1/2.Method 2. If the trig equation contains many trig functions, transform it into an equation that contains only one trig function as a variable.Example 5. Solve 3cos ^2 x - 2sin^2 x = 1 - 3sin xTransform the equation into the one containing only sin x. Replace cos^2 x = 1 - sin^2 x (Trig Identity 1). Call sin x = t.3(1 - sin^2 x) - 2sin^2x +3sin x - 1 = 03 - 3t^2 - 2t^2 + 3t - 1 = -5t^2 + 3t + 2 = 0.This is a quadratic equation with 2 real roots 1 and -2/5. Next solve the 2 trig basic equations: sin x = t = 1 and sin x = t = -2/5.The common period of a trig equation.The common period of a given trig equation must equal the least multiple of all the contained trig functions' periods.Example: The equation cos x + tan x = 1 has 2Pi as common period.The equation f(x) = sin 2x + cos x = 0 has 2Pi as common periodThe equation sin x + cos x/2 has 4 Pi as common period.Unless specified, a trig equation must be solved covering at least one common period.Solving special types of trig equations.There are a few special types of trig equations that require specific transformations.Examples: asin x + bcox x = ca(sin x + cos x) +bsin x.cos x = casin^2 x + bsin x.cos x + c cos^2 x = 0.Checking answers.Solving trig equation is a tricky work that easily leads to errors and mistakes. The answers should be carefully checked.After solving, you may check the answers by using graphing calculators. To know how, see the book mentioned above.(This article was written by Nghi H. Nguyen, the co-author of the new Diagonal Sum Method for solving quadratic equations)
Is there an analytical way to solve trig functions?
yes. there are certain formulae for solving them. dont remember
When do weather men use algebra 2?
We don't. We then learn trig, calculus, and then differential equations, and we use that.
Are trigonometry and algebra 2 the same?
No, not even close. Though both work with variables in some instances and other mathematical techniques, such as logarithms and algebraic manipulation, algebra is mostly equation solving to get the variables value, though trig has equations, while trig is the study of triangular and circular measurement and using these measurements to solve specific problems. Trig is much about identities, functions and many formulas while algebra is mostly about function and equation manipulation. Still, they are both mathematical disciplines.
Trigonometry equations used in everyday life?
Your brain is constantly doing trig subconsciously when you see(unless you have only 1 working eye). When we see things to gain depth perception, and thus gauge distance that is trig.
Who are the important people in trigonometry?
The students and teacher of my trig. class
What is the period of a trig function?
The period of trigonometric function is the distance between repetitions of the function. The "x" value of the space it takes to start over.
What is the birth name of Trig Palin?
Trig Palin's birth name is Trig Paxson Van Palin.
Is the avd algreba or algebra trig is harder?
trig
What is elective math?
Students who have successfully completed Algebra II may take this year long elective course that extends both algebra and trigonometry topics. Emphaisis is on functions and their inverses, equations and inequalities, graph theory, matrices, sequences and series, conic sections, and logarithms. Trigonometry topics include triangular and circular definitions of the trig functions, trig identities, and solutions of trig equations. This course is designed for students who do not plan to continue with math studies in high school but needed to strengthen and extend their foundations in algebra and trigonometry. It does not meet the prerequisites for Calculus AB and BC.
Is Trig Palin still in Afghanistan?
No Trig is not in Afghanistan cause he is to young to serve. Trig is Palin's youngest child.
