Question 1046316
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solve 3 sin^2 theta=4cos theta-1
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<pre>
{{{3*sin^2(theta)}}} = {{{4*cos(theta)-1}}}

Replace {{{sin^2(theta)}}} by {{{1-cos^2(theta)}}} to get the equation uniform for {{{cos(theta)}}}:

{{{3(1-cos^2(theta))}}} = {{{4*cos(theta)-1}}},

{{{3-3*cos^2(theta)}}} = {{{4*cos(theta)-1}}},

{{{3cos^2(theta) + 4cos(theta) - 4}}} = {{{0}}}.


Factor left side:


(3cos(x)-2)*(cos(x)+2) = 0.


The equation deploys in two independent equations


1.  {{{3cos(theta) - 2}}} = 0  --->  {{{cos(theta)}}} = {{{2/3}}}  --->  {{{theta}}} = +/-{{{arccos(2/3) + 2k*pi}}},  where k is any integer,  and


2.  {{{cos(theta) + 2}}} = 0  --->  {{{cos(theta)}}} = -2,  which has no solutions.


<U>Answer</U>.  The solutions are  {{{theta}}} = +/-{{{arccos(2/3) + 2k*pi}}},  where k is any integer.
</pre>

<TABLE> 
  <TR>
  <TD> 

{{{graph( 330, 330, -6.5, 6.5, -8.5, 8.5,
          3*(sin(x))^2-4*cos(x)+1
)}}}


Plot y = {{{3*sin^2(x)-4cos(x)+1}}}

  </TD>
  </TR>
</TABLE>