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

Algebra ->  Trigonometry-basics -> 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      Log On


   

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) About Me  (Show Source):
You can put this solution on YOUR website!
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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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Since the three terms (1/24)*sin(A), (1/3), and tan(A) form a GP, it implies that


    tan%28A%29%2F%28%281%2F3%29%29 = %28%281%2F3%29%29%2F%28%281%2F24%29%2Asin%28A%29%29


and hence


    %281%2F24%29%2Asin%28A%29%2Atan%28A%29 = 1%2F9

    %281%2F8%29%2A%28sin%5E2%28A%29%2Fcos%28A%29%29 = 1%2F3

    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%5B1%2C2%5D = %28-8+%2B-+sqrt%28%28-8%29%5E2+%2B+4%2A3%2A3%29%29%2F%282%2A3%29 = %28-8+%2B-+sqrt%28100%29%29%2F6 = %28-8+%2B-+10%29%2F6.


So, one root is  x%5B1%5D = -8+%2B+10%29%2F6 = 2%2F6 = 1%2F3,  and it implies   cos(A) = 1%2F3.



Another root is  x%5B2%5D = -8+-+10%29%2F6 = -18%2F6 = -3,  and it does not produce the corresponding cosine.


ANSWER.  Under the given conditions,  cos(A) = 1%2F3.

Solved (without using a calculator, as requested).