How to solve sin3x<sinx?

2 Answers
Jun 13, 2018

The solution is x in (pi/4, 3/4pi)uu(pi,5/4pi)uu(7/4pi, 2pi), [2pi]

Explanation:

To solve this trigonometric inequality, we need

sin3x=3sinx-4sin^3x

The inequality is

sin3x<sinx

=>, sin3x-sinx<0

Let f(x)=sin3x-sinx

The period of f(x) is T=2pi

Study the functio on the interval I= [0, 2pi]

Therefore,

f(x)=3sinx-4sin^3x-sinx

=2sinx-4sin^3x

=2sinx(1-sin^2x)

Let 2sinx(1-sin^2x)=0

The solutions to this equation are

{(sinx=0),(1-2sin^2x=0):}

<=>, {(x=0+kpi),(x=pi/4+kpi), (x=5/4pi+kpi):}

Build a sign chart

color(white)(aaaa)xcolor(white)(aaaa)0color(white)(aaaa)pi/4color(white)(aaaa)3/4picolor(white)(aaaa)picolor(white)(aaaa)5/4picolor(white)(aaaa)7/4picolor(white)(aaaa)2pi

color(white)(aaaa)sinxcolor(white)(aaaa)+color(white)(aaaa)+color(white)(aaaa)+color(white)(aaa)-color(white)(aaaa)-color(white)(aaaa)-

color(white)(a)1-2sin^2xcolor(white)(aa)+color(white)(aaaa)-color(white)(aaaa)+color(white)(aaa)+color(white)(aaaa)-color(white)(aaaa)+

color(white)(aaaa)f(x)color(white)(aaaa)+color(white)(aaaa)-color(white)(aaaa)+color(white)(aaa)-color(white)(aaaa)+color(white)(aaaa)-

Therefore,

f(x)<0 when x in (pi/4, 3/4pi)uu(pi,5/4pi)uu(7/4pi, 2pi), [2pi]

graph{sin(3x)-sinx [-8.54, 37.08, -9.16, 13.64]}

Jun 13, 2018

sin 3x - sin x < 0
f(x) = 2cos (2x). sin x < 0
First solve the 2 basic trig equations to find the end-points (critical points)
1. g(x) = cos 2x = 0 --> 2x = pi/2, and 2x = (3pi)/2 -->
x = pi/4; x = (3pi)/4; x = (5pi)/4, and x = (7pi)/4
2. h(x) = sin x = 0 --> x = 0; x = pi; and x = 2pi
Next, to algebraically solve the trig inequality, create a sign chart that shows the variation of the 2 functions g(x) and h(x) when x varies from 0 to 2pi through the critical points.
(0, pi/4, (3pi)/4, pi, (5pi)/4, (7pi)/4, 2pi)
The sign (+ or -) of the function f(x) is the resulting sign of the 2 function g(x) and h(x)
The solutions are the intervals
(pi/4, (3pi)/4) where g(x) < 0, and h(x) > 0 and --> f(x) < 0
(pi, (5pi)/4) where g(x) > 0, and h(x) < 0, and --> f(x) < 0
((7pi)/4, 2pi) where g(x) > 0, and h(x) < 0 , and --> f(x) < 0
You also can solve the trig inequality by using graphing calculator.
The parts of the graph that lie below the x-axis represent the answers inside the interval (0, 2pi)
Note. The graph shows x in radians. Exp. x = pi = 3.14;
x = pi/2 = 1.57