SOLVING TRIG INEQUALITIES Compare the current sign chart method to the Nghi Nguyen method
GENERALITIES.
To solve a complex trig inequality we usually transform it into a product of basic trig
inequalities in the form:
F(x) = f(x).g (x) ≤ 0 (or ≥) or F(x) = f(x).g(x).h(x) ≤ 0 (or ≥ 0)
There are, so far, 2 main methods to solve complex trig inequalities.
1. THE ALGEBRAIC METHOD USING THE SIGN CHART
There are, in general, 4 steps in solving a complex trig inequalities by using the sign
chart.
Step 1. Transform the inequality into standard form.
Example. The inequality (cos 2x ≤ 2 + 3sin x) will be transformed into standard form:
F(x) = cos 2x – 3sin x – 2 ≤ 0
Step 2. Find the common period
The common period of the complex trig inequality is the least common multiple of all
the periods presented in the inequality.
Examples: The common period of
F(x) = sin x + cos 2x + sin 3x is: 2Ꙥ
F(x) = tan x + 2cot x is: Ꙥ
F(x) = sin 2x + 3cos 2x + sin (x/2) is: 4Ꙥ
Step 3. Transform the given trig inequality F(x) into a product of one or many basic trig
inequalities:
a. If F(x) contains only one trig function, or similar, solve it as a basic trig inequality.
Example.
F(x) = sin (x – 135⁰) < √3/2
F(x) = tan (2x – Ꙥ/3) > √2
b. If the complex inequality F(x) contains 2 or more trig functions, transform F(x) into
one or many basic trig inequalities (or similar) in the form:
F(x) = f(x).g(x) ≤ 0 (or ≥ 0)
or
F(x) = f(x).g(x).h(x) ≥0 (or ≤ 0)
Page 1 of 4
To know how to transform F(x) into 2 or 3 basic trig functions, please read the article
titled “Solving trig equations – Concept and method” by the same author.
Examples of complex trig inequalities:
F(x) = sin x + sin 3x > 1
F(x) = sin (x/2).cos (x – Ꙥ/3) > 0
F(x) = cos 2x + sin 2x < cos x
F(x) = tan 2x.tan x < 1
STEP 4. Creating the Sign Chart
After transformation, we get F(x) = f(x).g(x) ≤ 0 (or ≥ 0)
First, solve f(x) = 0 to find its real roots in x.
Plot these x- values on the first line of a sign chart with x progressively increases
inside the common period limit. These values of x create various colons (called
intervals) on the sign chart.
Find and mark the sign status (+ or -) of the trig function f(x) inside each interval on the
second line of the chart.
Do the same thing for the trig function g(x) on the third line of the chart.
On the last line, the resulting sign of F(x) = f(x).g(x) will show the combined solution
set of the trig inequality F(x).
Example.
Solve
sin x + sin 3x < - sin 2x
Solution. First step → Standard form: F(x) = sin x + sin 2 x + sin 3 x < 0
Second step. Common period → 2Ꙥ
Third step. Use trig identity (sin a + sin b) to transform the inequality F(x).
F(x) = sin x + sin 3x + sin 2x = 2sin 2x.cos x + sin 2x = sin 2x(2cos x+ 1) < 0. We have
the form: F(x) = f(x).g(x) < 0
1. Solve f(x) = sin 2x = 0 → there are 4 real roots at: (0), (Ꙥ/2), (Ꙥ) and (3Ꙥ/2)
Plot these 4 values on the first line of the sign chart inside the period (0, 2Ꙥ).
Find the sign status of f(x) = sin 2x when x varies inside these 4 intervals.
The function f(x) = sin 2x > 0 when x varies inside the intervals: (0, Ꙥ/2). Find and
mark the sign status (+ or -) of f(x) inside the 3 other intervals.
2. Next solve g(x) = 2cos x + 1 = 0. This gives: cos x = - 1/2 = cos (± 2Ꙥ/3)
There are 2 real roots at: x = 2Ꙥ/3, and x = 4Ꙥ/3. Plot these 2 x-values in their
position on the first line. Select the check point (Ꙥ).
We get g(Ꙥ) = - 2 + 1 = - 1 < 0. The function g(x) is negative (< 0) inside the interval
(2Ꙥ/3, 4Ꙥ/3). Mark the sign status of g(x) on the line 3 of the chart.
Page 2 of 4
Finally, create the combined sign chart of F(x) = f(x).g(x)
x |0
Ꙥ/2
2Ꙥ/3
Ꙥ
4Ꙥ/3
3Ꙥ/2
2Ꙥ
-------------------------------------------------------------------------------------------------------f(x) |0
+
0
|
- 0
+
|
+
0
0
-------------------------------------------------------------------------------------------------------g(x) |
+
|
+
0 |
0
+
|
+
|
-------------------------------------------------------------------------------------------------------F(x) |
| yes
|
|
yes |
|
yes
|
The combined solution set of F(x) < 0 are the 3 open intervals
(Ꙥ/2, 2Ꙥ/3) and (Ꙥ, 4Ꙥ/3) and (3Ꙥ/2, 2Ꙥ)
THE NGHI NGUYEN METHOD.
This innovative method (Google, Bing, Yahoo Search), visually solves complex trig
inequalities by using the number unit circle, that is numbered in radians or degrees.
Its 4 steps are exactly the same as the sign chart method
First step. Bring the complex inequality to the standard form
Example. sin x + sin 2x < - sin 3x.
Standard form: F(x) = sin x + sin 2x + sin 3x < 0
Second step. Find common period. In this example: 2Ꙥ
Third step. Transform the inequality F(x) into the form: f(x).g(x) ≤ 0 (or ≥ 0).
F(x) = sin x + sin 3x + sin 2x = 2sin 2x.cos x + sin 2x = sin 2x(2cos x + 1) < 0
Fourth step. Solve separately the 2 basic inequalities by the number unit circle.
1. Solve f(x) = sin 2x = 0. This give 3 solutions
sin 2x = 0. --→ 2x = 0 + 2kꙤ --→ x = 0 + kꙤ
sin 2x = Ꙥ --→ 2x = Ꙥ + 2kꙤ --→ x = Ꙥ/2 + kꙤ
sin 2x = 2Ꙥ --→ 2x = 2Ꙥ + 2kꙤ --→ x = Ꙥ + kꙤ
For k = 0, and k = 1, there are 4 end points at : 0, Ꙥ/2, Ꙥ, and 3Ꙥ/2. Plot these end
points on a unit circle. They divide the unit circle into 4 arc lengths. Find the sign status
of f(x) by selecting the check point (x = Ꙥ/4). We get f(Ꙥ/4) = sin (Ꙥ/2) = 1 > 0.
Therefore, f(x) is positive (> 0) inside the arc length (0, Ꙥ/2). Color the 3 other
intervals (blue, red, blue).
Page 3 of 4
2. Solve g(x) = (2cos x + 1) = 0
This give cos x = - 1/2 = cos (± 2Ꙥ/3). There are 2 end points at (2Ꙥ/3) and (4Ꙥ/3),
and two arc lengths.
Find the sign status of g(x) by selecting point (Ꙥ) as check point. We get g(Ꙥ) = - 2 +
1 = - 1 < 0. Then, g(x) is negative (< 0) inside the interval (2Ꙥ3, 4Ꙥ/3). Color it blue
and color the rest red.
By superimposing, the combined solution set of F(x) < 0 are the 3 open intervals:
(Ꙥ/2, 2Ꙥ/3) and (Ꙥ, 4Ꙥ/3) and (3Ꙥ/2, 2Ꙥ). Same answer as the sign chart method.
THE ADVANTAGES OF THE NGHI NGUYEN METHOD
1. In the sign chart method, the two extremities of the sign chart are separate. In the
new method, on the unit circle, these two extremities joint together and show the
periodic character of trig functions.
2. In the sign chart method, all values of x from all operations are progressively placed
on the first line only. In case of complex trig inequality like F(x) = f(x).g(x).h(x) ≥ 0
(or ≤ 0), the high number of x-values make the chart stuffy and confusing. In the new
method, the number of x depends only on, in each step, the solving of one basic trig
inequality.
3. The simple rules of end points and arc lengths make the new method more
convenient than the sign chart method.
(This article was authored by Nghi H Nguyen on Feb. 22, 2021)
Page 4 of 4