Question

Find all solutions to sin x = 1/2√3 tan x in the interval [ 0 , 2pi ).

How should I start to create like trig functions ?

Answer 1

substitute
\[\tan(x)=\frac{\sin(x)}{\cos(x)}\]

Answer 2

let me know if you need another step

Answer 3

Do I move the sin x to the right side by subtracting?

Answer 4

yes. then try to factor and use the ZPP(zero product property)

Answer 5

do you mean this
\[ \sin x =\frac{1}{2\sqrt{3}} \tan x\] or \[\sin x =\frac{1}{2\sqrt{3} \tan x}\]

Answer 6

good question. not sure. my answer has been assuming the first version you gave amir.

Answer 7

It's the first version.

@matheducatorMcG , is the cosine supposed to be 1/ cos x when you factored it & its on the demoninator? or it doesnt matter ?

Answer 8

this is what i meant
\[0=\sin(x)(\frac{1}{2\sqrt{3}\cos(x)}-1)\]

Answer 9

that second factor seems suspect.
\[\frac{1}{2\sqrt{3}\cos(x)}-1=0\]
\[\frac{1}{2\sqrt{3}\cos(x)}=1\]
\[1=2\sqrt{3}\cos(x)\]
\[\frac{1}{2\sqrt{3}}=cos(x)\]
I don't know that one off the top of my head. i would look at the graph of cos(x) and find intersections with
\[y=\frac{1}{2\sqrt{3}}\]

Answer 10

2 intersections at
\[x \approx1.2779536\] and \[x \approx 5.0052318\]

Answer 11

the other solutions for the sin(x)=0 were at 0, pi, and 2pi