Lesson Solving trigonometric inequalities

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GENERALITIES

Definition.

A trig inequality is an inequality in the form of F(x) < 0 (or > 0), in which F(x) contains one or a few trig functions of the variable arc x.
Solving the inequality F(x) means finding all the values of the variable arc x, within the common period, that make the inequality true. These values of x constitute the solution set of the trig inequality.
Solution sets of trig inequalities are expressed in the form of intervals.

Examples of trig inequalities: sin (x + Pi/3) < 0.5 ; cos ( 2x - Pi/5) < -1 ; sin 2x + cos x > 0.5 ; sin x + sin 2x < -sin 3x  ; tan x + cot x < -3 ;

Examples of solution sets of trig inequalities: (Pi/3, 2Pi/3);[0, 2Pi];(30 degree, 145 degree);
(-Pi/2, Pi/2)

COMMON PERIOD.

The common period of a trig inequality should be the least multiple of all periods of the trig functions presented in the trig inequality. Examples:
The trig inequality sin x + cos 2x < 0 has 2Pi as common period.
The trig inequality sin x + sin 2x + cos x/2 < 0.5 has 4Pi as common period.
The trig inequality tan x + cot 2x > 3 has Pi as common period.
Unless specified, the solution set of a trig inequality must be solved covering a common period.

THE TRIG UNIT CIRCLE.

It is a circle with radius R = 1 unit. The variable arc AM, that rotates counter-clockwise on the trig unit circle, defines 4 common trig functions of the arc x.
The horizontal axis OAx defines the function f(x) = cos x
The vertical axis OBy defines the function f(x) = sin x.
The tangent vertical axis AT defines the function f(x) = tan x.
The tangent horizontal axis BU defines the function f(x) = cot x

The trig unit circle is also used as proof to solve simplest trig inequalities.

Examples:
For the trig inequality sin x > 0.5, the trig circle shows the answer when the arc x varies between the values Pi/6 and 5Pi/6; answer: Pi/6 < x < 5Pi/6.
For the trig inequality tan x > 1, the trig circle shows the answer when the arc x varies between the values Pi/4 and Pi/2 and between 5Pi/4 and 3Pi/2. 
Answers: Pi/4 < x < Pi/2 and 5Pi/4 < x < 3Pi/2.

BASIC TRIG INEQUALITIES.

They are also called trig inequalities in simplest forms. There are 4 common basic trig inequalities:

sin x < (or >) a  ;  cos x < (or >) a  ;  tan x < (or >) a  ;  cot x < (or > 0) a.

Solving basic trig inequalities proceeds, first by using trig conversion table (or calculator), then next, by considering the various positions of the arc x that rotates counter-clockwise on the trig unit circle.

Example 1. Solve sin x > 0.709.
Solution. The answers are given by calculator and trig circle:
Pi/4 < x < 3Pi/4  ;  (Answer within 2Pi period).
Pi/4 + 2k.Pi < x < 3Pi/4 + 2k.Pi  ;  (Extended answer).

Example 2. Solve: tan x > 0.414 (within period 2Pi).
Solution. The answers are given by the trig circle and calculator:
Pi/8 < x < Pi/2
5Pi/8 < x < 3Pi/2

Example 3. Solve: cos (2x + 45 deg.) < 0.5 (within period 360 deg.)
Solution. Answers given by calculator and trig circle:
60 deg. < (2x + 45 deg.) < 300 deg.
15 deg. < 2x < 255 deg.
7.5 deg. < x < 125 deg.

STEPS IN SOLVING TRIG INEQUALITIES.

In general, there are 4 steps in solving trig inequalities.

STEP 1. Transform the given trig inequality into standard form F(x) < 0 (or > 0). Examples:
The trig inequality (cos 2x < 2 + 3sin x) will be transformed into F(x) = cos 2x - 3sin x -2 < 0.
The trig inequality (sin x + sin 2x < -sin 3x) will be transformed into F(x) = sin x + sin 2x + sin 3x < 0.

STEP 2. FIND THE COMMON PERIOD.

The common period of a trig inequality must be the least multiple of all period of the trig functions presented in the inequality. Examples:

The inequality F(x) = sin x + cos 2x - sin x/2 > 0.5 has 4Pi as common period.
The inequality F(x) = tan x - cot x/2 < 1 has 2Pi as common period.

STEP 3. SOLVE THE TRIG EQUATION F(x) = 0.

- If the given trig inequality contains only one trig function, solve it as  basic trig equations.

Example 4. Solve: F(x) = sin^2 x + 3sin x - 4 = 0.
Solution. Call sin x = t. The equation becomes: t^2 + 3t - 4 = 0. It is a quadratic equation that has 2 real roots t = 1 and t = -4. Next solve the basic inequality t = sin x = 1. The other real root sin x = t = -4 is rejected since it is < -1.

- If the given trig inequality contains two or more trig functions, there are two methods to solve F(x) = 0, depending on transformation possibilities.

1. METHOD 1. Transform F(x) into a product of many basic trig inequalities. Next, solve these basic inequalities to get all values of x within the common period. These x values will be used in Step 4.

For transformation means, use common algebraic transformation (common factor, polynomial properties,...), definitions and properties of trig functions, and trig identities (the most needed).

Example 5. Solve: sin 2x + 2cos x = 0. 
Solution. Using trig identity, substitute  sin 2x by 2sin x.cos x in the equation.
2sin x.cos x + 2cos x = 0 = 2cos x(sin x + 1) = 0. 
Next, solve the 2 basic trig inequalities: cos x = 0 and sin x = -1.

Example 6. Solve: cos x + cos 2x + cos 3x = 0.
Solution. First, transform the sum (cox + cos 3x).
(cos x + cos 3x) + cos 2x = 2cos 2x cos x + cos 2x = cos 2x(2cos x + 1) = 0
Next, solve the 2 basic trig inequalities: cos 2x = 0 and cos x = -1/2.

Example 7. Solve: sin x - sin 3x = cos 2x.
Solution. Using trig identity to transform the sum (sin x - sin 3x).
(sin x - sin 3x) - cos 2x = 2cos 2x.sin x - cos 2x = cos 2x(2sin x - 1) = 0.
Next, solve the 2 basic trig inequalities: cos 2x = o and sin x = 1/2.

2. METHOD 2. If the given trig inequality contains 2 or more trig functions, transform it into a trig inequality containing only one trig function. There are a few Tips on how to select the appropriate 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 8. Solve 4sin^2 x + 3cos x + 3 = 0. 
Solution. Select cos x = t. Substitute sin^2 x in the inequality by (1 - cos^2 x) = 1 - t^2. We get:
4(1 - t^2) + 3t + 3 = -4t^2 + 3t + 7 = 0. It is a quadratic equation having 2 real roots: -1 and  
7/4. 
Next solve the 2 basic trig inequalities: cos x = t = -1. The other real root is rejected since > 1.

Example 8. Solve: tan x + 2tan^2 x - cot x - 2 = 0.
Solution. Select t = tan x. The inequality becomes:
t + 2t^2 - 1/t - 2 = t^2 + 2t^3 -1 - 2t = 0
t^2(1 + 2t) - 1 - 2t = (1 + 2t)(t^2 - 1) = 0.
Next, solve the 2 basic trig inequalities: (t = tan x = -1/2) and (t^2 = tan^2 x = 1), to get the x-values within the common period.

Example 9. Solve: cos 2x - 3sin x - 2 = 0.
Solution. Call sin x = t. Substitute cos 2x by (1 - 2sin^2 x) = 1 - 2t^2 into the inequality:
(1 - 2t^2 - 3t -2 = -2t^2 - 3t - 1 = 0. It is a quadratic equation that has 2 real roots: -1 and -1/2. Next solve the 2 basic trig inequalities: six = t = -1 and sin x = t = -1/2, to get all values of x within the common period.

STEP 4. SOLVE THE TRIG INEQUALITY F(x) < (or >) 0  BY THE ALGEBRAIC APPROACH.

Based on the values of x, obtained from Step 3, within the common period of the given inequality, we create a Sign Table. Suppose the trig inequality had been transformed into 2 basic trig functions: F(x) = g(x).h(x) < (or >) 0.

The first line of the Sign Table figures all consecutive values of x (from step 3), going increasingly within the common period. These values of x create various intervals between them.
The second line of the Sign Table, figures the real roots of g(x) = 0  and in the same time, the variation of g(x) with its positive and negative values between corresponding intervals. This variation of g(x) is obtained by considering various positions of the arc x on the trig unit circle. 
The third line figures the real roots of h(x) = 0, and in the same time the variation of h(x) with its negative and positive values in corresponding intervals.
The last line figures all the real roots of F(x) = 0 and the variation of F(x). The Sign of F(x) in each interval is the combined sign of the product: g(x).h(x). The solution set of F(x) can be easily seen on the Sign Table.

Example 10. Solve: sin x + sin 3x < -sin x.
Solution. Step 1, write the inequality in standard form: F(x) = sin x + sin 2x + sin 3x < 0.
Step 2: the common period is 2Pi.
Step 3: Solve F(x) = (sin x + sin 3x) + sin 2x = 2sin 2x.cos x + sin 2x = sin 2x(2cos x + 1) = 0
Next, solve g(x) = sin 2x. The solution arcs are: 0; Pi/2; Pi; 3Pi/2; 2Pi.
Then, solve h(x) = 2cos x + 1 = 0. The solution arcs are: 2Pi/3 and 4Pi/3.

Create a Sign Table with all values of x from Step 3 within the common period from 0 to 2Pi.

x ...... 0 ...... Pi/2 ....... 2Pi/3 ....... Pi ....... 4Pi/3 ....... 3Pi/2 ....... 2Pi.

g(x) .. 0 ... + .... 0 .. - ............ - .... 0 ... + .......... + ...... 0 .... - .... 0

h(x) .. 0 ... + ......... + ...... 0 .... - ......... - .... 0 .... + .............. + .....

F(x) .. 0 ... + .. 0 .. - ...... 0 .... + ... 0 ... - .... 0 ... + ..... 0 .... - ..... 0

We see that F(x) is negative (< 0) within the intervals: (Pi/2, 2Pi/3) and (Pi, 4Pi/3) and (3Pi/2, 2Pi). That is the solution set of the trig inequality F(x) < 0.

NOTE 1. The approach to determine the variation of g(x) and h(x), within the common period, is exactly the same approach in solving basic trig inequalities by considering the various positions of the variable arc that rotates on the trig unit circle.
Example: Find the variation of g(x) = sin x - 0.5 > 0, within period 2Pi. The calculator and the trig unit circle shows the answer when Pi/6 < x < 5Pi/6. Within this interval, g(x) is positive (> 0). In the remainder part of the circle, g(x) < 0.

NOTE 2. THE GRAPHING METHOD.

a. After solving, students can use graphing calculators to check the real roots of the trig equation F(x) = 0. These root values will be given in decimals. For examples, Pi is given as 3.14; 360 degree is given as 6.28.
b. Solving trig inequalities is a tricky work that often leads to errors and mistakes. Students can  use graphing calculators to check the answers after solving trig inequalities by the algebraic method, described above. 
c. Students can also use graphing calculator to directly solve the given trig inequality F(x) < 0 (or > 0), by considering the position of the graph of F(x) over the x-axis in various intervals within the common period. This method, if allowed by tests/exams, is fast, accurate and convenient. To know more about the graphing method, see book titled: "Trigonometry: Solving trigonometric equations and inequalities" (Amazon e-book 2010).