Question 1063200
.
I need to solve this equation for 0 < x < 2pi
 -2tan(-2x + 4pi/3) = 2sqrt3 
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


<pre>
{{{ -2tan(-2x + 4pi/3)}}} = {{{2*sqrt(3)}}}  ----->  (divide both sides by (-2) )  --->

{{{tan(-2x+ 4pi/3)}}} = {{{-sqrt(3)}}}  ---->  

The last equation has solutions  {{{-2x + 4pi/3}}} = {{{2pi/3 + k*pi}}}, k = 0, +/-1, +/-2, , , , .


Let's derive an expression for x from this:
 

{{{-2x + 4pi/3}}} = {{{2pi/3 + k*pi}}}   ---->  {{{2*(-x + 2pi/3)}}} = {{{2pi/3 + k*pi}}}  ---->  {{{-x + 2pi/3}}} = {{{pi/3 + k*(pi/2)}}}  ---->  x = {{{pi/3 + k*(pi/2)}}}, k = 0, +/-1, +/-2, , , , .

In the given interval ({{{0}}},{{{2*pi}}}) 

x = {{{pi/3}}}, {{{pi/3 + pi/2}}}, {{{pi/3+pi}}}, {{{pi/3+ 3pi/2}}} are the solutions.

<U>Answer</U>. In the given interval the solutions are   = {{{pi/3}}}, {{{5pi/6}}}, {{{4pi/3}}}, {{{11pi/6}}}.    (See the plot below)
</pre>

{{{graph( 330, 330, -1.5, 6.5, -2.5, 4.5,
          -2tan(-2x + 4pi/3), 2*sqrt(3)
)}}}


Plots y = {{{-2tan(-2x + 4pi/3)}}} and y = {{{2*sqrt(3)}}}


To see more examples of solved trigonometry equations with detailed solutions, look into the lessons 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solving-simple-problems-on-trigonometric-equations.lesson>Solving simple problems on trigonometric equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solving-typical-problems-on-trigonometric-equations.lesson>Solving typical problems on trigonometric equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solving-more-complicated-problems-on-trigonometric-equations.lesson>Solving more complicated problems on trigonometric equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Solved-problems-on-trigonometric-equations.lesson>Solving advanced problems on trigonometric equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/Proving-Trigonometry-identities.lesson>Proving Trigonometry identities</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Trigonometry-basics/OVERVIEW-of-lessons-on-calculating-trig-functions-and-solving-trig-equations.lesson>OVERVIEW of lessons on calculating trig functions and solving trig equations</A>

in this site.



Also, you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this online textbook under the topic "<U>Trigonometry: Solved problems</U>".