Question 1177382
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The three numbers (1/24)*sin(A), (1/3), and tan(A) are in geometric progression. 
Find the numerical value of cos(A), where 0 degrees < A < 90 degrees. Should be solved without the use of a calculator.
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<pre>
Since the three terms (1/24)*sin(A), (1/3), and tan(A) form a GP, it implies that


    {{{tan(A)/((1/3))}}} = {{{((1/3))/((1/24)*sin(A))}}}


and hence


    {{{(1/24)*sin(A)*tan(A)}}} = {{{1/9}}}

    {{{(1/8)*(sin^2(A)/cos(A))}}} = {{{1/3}}}

    3*(1-cos^2(A)) = 8*cos(A)


Introduce new variable  x = cos(A)  and write the last equation in the form


    3 - 3x^2 = 8x

    3x^2 + 8x - 3 = 0

    {{{x[1,2]}}} = {{{(-8 +- sqrt((-8)^2 + 4*3*3))/(2*3)}}} = {{{(-8 +- sqrt(100))/6}}} = {{{(-8 +- 10)/6}}}.


So, one root is  {{{x[1]}}} = {{{-8 + 10)/6}}} = {{{2/6}}} = {{{1/3}}},  and it implies   cos(A) = {{{1/3}}}.



Another root is  {{{x[2]}}} = {{{-8 - 10)/6}}} = {{{-18/6}}} = -3,  and it does not produce the corresponding cosine.


<U>ANSWER</U>.  Under the given conditions,  cos(A) = {{{1/3}}}.
</pre>

Solved (without using a calculator, as requested).