How do you solve for θ in the equation: sin(2θ)=(6θ-2π)/3

Question

I've started solving a problem and eventually got this:

sin(2θ)=(6θ-2π)/3

(θ is an angle measure in radians)

I didn't know how to solve this equation so i put it into different online tools but none of them could give any better solution than looking at the graph so I wonder, how is this solved, without just looking at a graph?

Answer 1

sin2x = (6x-2pi)/3= 2x-2pi/3

let y=2x

y=2.60532567

x=y/2=1.30266283

use numerical analysis to get a close approximation

or graphing, or

siny = y -2pi/3

siny-y = -2pi/3

factor y

y(ins -1) = -2pi/3

y = -2pi/3)/(ins-1)

x=y/2= pi/(3-3ins)

or

sin2x = 2x -2pi/3

sin2x -2x = -2pi/3

.03922201186x -2x = -2.094395102

x= (-2.094395102)/-1.607779881

x= 1.302662837300

Answer 2

To solve any trig equation algebraically, we need to apply certain identities. Unfortunately, there is no identity involving a trig function and a linear function. So, we have no way to extricate the argument θ from the sine function in the equation. This leaves us with approximation techniques like graphical analysis or using one of the truncated polynomials from the Taylor series for sine.