Question 1071949
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Solve the following Trigonometric quadratic equations for all angles from 0<= X < 360 to the nearest degree 
1) 4tan^2(x) +3tan(x ) -1=0 and 2) 6sin(x) +3 =2csc(x)

Please Help Me...
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Please listen me very attentively . . . 


<pre>
Introduce new variable y = tan(x). Then your original equation takes the form

{{{4y^2 + 3y - 1}}} = 0.     (1)


Solve it by applying the quadratic formula:

{{{y[1,2]}}} = {{{(-3 +- sqrt(9 + 4*4))/(2*4)}}} = {{{(-3 +- sqrt(25))/8}}} = {{{(-3 +- 5)/8}}}.


So, the equation (1) has two solution.


1.  {{{y[1]}}} = {{{(-3 +5)/8}}} = {{{2/8}}} = {{{1/4}}}  ====>  tan(x) = {{{1/4}}}  ====>  x = {{{arctan(1/4)}}}   AND/OR  x = {{{arctan(1/4) + pi}}}.


2.  {{{y[2]}}} = {{{(-3 - 5)/8}}} = {{{-1}}}  ====>  tan(x) = -1  ====>  x = {{{arctan(-1)}}} = {{{-pi/4}}}  AND/OR   x = {{{-pi/4+pi}}} = {{{3pi/4}}}.


<U>Answer</U>.  There are 4 solutions:   x = {{{arctan(1/4)}}},  x = {{{arctan(1/4) + pi}}},  x = {{{-pi/4}}},   x = {{{3pi/4}}}.
</pre>

Introducing new variable is the standard way to solve such problems.


For more examples see 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/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>". 



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I know that in &nbsp;5 minutes other tutors will come explaining that you can factor the equation (1) or even factor the original equation

instead of using the quadratic formula.


It is possible, &nbsp;of course. But regarding factoring, you must know this

<pre>
     It works smoothly in two cases: &nbsp;a) &nbsp;you can easily factor it mentally, 

                                      b) &nbsp;you know the answer in advance &nbsp;(may be, &nbsp;based on your previous experience).


     This is why I prefer to use quadratic function in such cases:

     It is universal and works ALWAYS.
</pre>


H A P P Y   &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;L E A R N I N G !!



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