Question

How to solve `sin3x<sinx`?

Answer 1

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)` `x` `color(white)(aaaa)` `0` `color(white)(aaaa)` `pi/4` `color(white)(aaaa)` `3/4pi` `color(white)(aaaa)` `pi` `color(white)(aaaa)` `5/4pi` `color(white)(aaaa)` `7/4pi` `color(white)(aaaa)` `2pi`

`color(white)(aaaa)` `sinx` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-`

`color(white)(a)` `1-2sin^2x` `color(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]}

Answer 2

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