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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 Circle
It 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 x
The 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 period
The 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 = a
Solving 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 = 0.866
Solution. 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 = 0.5
Solution. 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 using
common 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 x
Transform 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 = 0
3 - 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 period
The 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 = c
a(sin x + cos x) +bsin x.cos x = c
asin^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)
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How do you solve trigonometric equations?
The answer depends on the nature of the equations.
Are trigonometric identities important?
Yes. Trigonometric identities are extremely important when solving calculus equations, especially while integrating.
How we solve equations of motion by graph?
One can solve equations of motion by graph by taking readings of the point of interception.
How do you solve imaginary equations?
The answer depends on the nature of the equation. Just as there are different ways of solving a linear equation with a real solution and a quadratic equation with real solutions, and other kinds of equations, there are different methods for solving different kinds of imaginary equations.
How can you solve problems involving equivalent expressions?
In the same way that you would solve equations because equivalent expressions are in effect equations
How do you solve trigonometric equations?
The answer depends on the nature of the equations.
Why do you solve trigonometric equations?
Use trigonometric identities to simplify the equation so that you have a simple trigonometric term on one side of the equation and a simple value of the other. Then use the appropriate inverse trigonometric or arc function.
What is the term trigonometry identity?
Trigonometric identities are trigonometric equations that are always true.
What are solutions of trigonometric equations?
That means the same as solutions of other types of equations: a number that, when you replace the variable by that number, will make the equation true.Note that many trigonometric equations have infinitely many solutions. This is a result of the trigonometric functions being periodic.
Are trigonometric identities important?
Yes. Trigonometric identities are extremely important when solving calculus equations, especially while integrating.
How do you find the locus of all points that lie on the graphs of both x plus 2y equals 1 and 2x -y equals 12?
You solve the two equations simultaneously. There are several ways to do it; one method is to solve the first equation for "x", then replace that in the second equation. This will give you a value for "y". After solving for "y", replace that in any of the two original equations, and solve the remaining equation for "x".You solve the two equations simultaneously. There are several ways to do it; one method is to solve the first equation for "x", then replace that in the second equation. This will give you a value for "y". After solving for "y", replace that in any of the two original equations, and solve the remaining equation for "x".You solve the two equations simultaneously. There are several ways to do it; one method is to solve the first equation for "x", then replace that in the second equation. This will give you a value for "y". After solving for "y", replace that in any of the two original equations, and solve the remaining equation for "x".You solve the two equations simultaneously. There are several ways to do it; one method is to solve the first equation for "x", then replace that in the second equation. This will give you a value for "y". After solving for "y", replace that in any of the two original equations, and solve the remaining equation for "x".
How can you solve this trigonometric bearing question?
Look there!
How do you solve tangent cosine tangent-cosine.?
It isn't clear what you want to solve for. To solve trigonometric equations, it often helps to convert other angular functions (tangent, cotangent, secant, cosecant) into the equivalent of sines and cosines. However, the details of course depend on the specific case.
How do you solve a variable?
You cannot solve a variable. You can solve an equation to find the value (or range of values) of a variable. How you do that depends on the nature of the equation that you have. Linear and quadratic equations are relatively simple, as are many trigonometric and exponential equations. But some cannot be solved in such a way and a numerical solution is required. Here you would make a guess and then improve on that guess and then improve on that until you were satisfied that you were close enough to the real answer.
How do you solve 9- 2 equations?
7
How do you solve easy trigonometric functions of angle?
With ease, I suppose. The question depends on what you consider easy trigonometric functions.
How do you solve these equations?
Tell me the equations first.
