SOLUTION: The three numbers (1/24)sinA, (1/3), and tan A are in geometric progression. Find the numerical value of cosA, where 0 degrees < A < 90 degrees. Should be solved without the use of

Question 1177382: The three numbers (1/24)sinA, (1/3), and tan A are in geometric progression. Find the numerical value of cosA, where 0 degrees < A < 90 degrees. Should be solved without the use of a calculator.
Answer by ikleyn(52863) (Show Source): You can put this solution on YOUR website!
.
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.
~~~~~~~~~~~~~

Since the three terms (1/24)*sin(A), (1/3), and tan(A) form a GP, it implies that


     = 


and hence


     = 

     = 

    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

     =  =  = .


So, one root is   =  =  = ,  and it implies   cos(A) = .



Another root is   =  =  = -3,  and it does not produce the corresponding cosine.


ANSWER.  Under the given conditions,  cos(A) = .

Solved (without using a calculator, as requested).

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