What are the ways on how to solve trigonometric equations?

Answer 1

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)