Question 1209808
.
Find the value of θ,
sin20°sinθ + sin100°sin(20 - θ)° = 0
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        In the post by  @CPhill, his solution,  giving the answer   {{{theta}}} = 40°,  is  INCORRECT.


<pre>
Let's check it.


We have  {{{sin(20^o)*sin(theta)}}} = sin(20°)*sin(40°) = 0.34202014332*0.64278760968 = 0.21984631.


Next, we have  

    {{{sin(100^o)*sin(20^o-theta)}}} = cos(10°)*sin(20°-40°) = cos(10°)*sin(-20°) = 0.98480775301*(-0.34202014332) = -0.336824089.


Thus 

    {{{sin(20^o)*sin(theta)}}} + {{{sin(100^o)*sin(20^o-theta)}}} = sin(20°)*sin(40°) + cos(10°)*sin(-20°) = 0.21984631 + (-0.336824089) = -0.116977779.


is not zero.
</pre>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Thus the answer by @CPhill is disproved.



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The right solution can be found using numerical methods.


I used a plotting tool in web-site  https:\\www.desmos.com/calculator/


It produces plots and is smart enough to make all necessary accompanying calculations automatically.


See my plot of participating functions in this web-page


<A href=https://www.desmos.com/calculator/da6cvuhjij>https://www.desmos.com/calculator/da6cvuhjij</A>


https://www.desmos.com/calculator/da6cvuhjij



Our solutions are the intersection points of the plots.


One intersection point is  &nbsp;x= 0.5236 radians, &nbsp;or &nbsp;30 degrees.


Another intersection point is &nbsp;x = 3.66519 radians, &nbsp;or &nbsp;210 degrees.


// To see the coordinates of the intersection points, &nbsp;click on these points.


So, &nbsp;numerically we get this answer: &nbsp;the angle &nbsp;{{{theta}}}  &nbsp;may have two values : &nbsp;30°  &nbsp;and/or  &nbsp;210°.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Now I will &nbsp;{{{highlight(highlight(PROVE))}}} &nbsp;to you &nbsp;{{{highlight(highlight(MATHEMATICALLY))}}} &nbsp;that these answers are correct.



<pre>
Indeed, for  {{{theta}}} = 30°

    {{{sin(20^o)*sin(theta)}}} = sin(20°)*sin(30°) = {{{(1/2)*sin(20^o)}}}   <<<---===  since  sin(30°) = {{{1/2}}}.


    {{{sin(100^o)*sin(20^o-theta)}}} = cos(10°)*sin(-10°) = {{{-(1/2)*cos(10^o)*sin(10^o)}}} = {{{-(1/2)*sin(20^o)}}}  <<<---=== sinse  sin(a)*cos(a) = {{{(1/2)*sin(2a)}}}.


Now you see that {{{theta}}} = 30° is the solution: it is PROVED.


Similar proof works for {{{theta}}} = 210°.
</pre>

At this point, &nbsp;the wrong solution of &nbsp;@CPhill is disproved completely,
and the right solution is found numerically and proved mathematically.