SOLUTION: What is the value of (sin 13° + sin 47° + sin 73° + sin 107°)/(cos 17°)?

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Question 1177720: What is the value of (sin 13° + sin 47° + sin 73° + sin 107°)/(cos 17°)?
Answer by ikleyn(52803) About Me  (Show Source):
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Compute     .

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                    Solution


Let simplify the numerator,  step by step. 


(1)  sin(73°) = cos(90° - 73°) = cos(17°).


     sin(107°) = sin(73°) = cos(17°)


     So, the sum of the third and the fourth terms in the numerator is  2*cos(17°).




(2)  sin(13°) + sin(47°) = ( use the basic formuls of trigonometry sin(a) + sin(b) = 2%2Asin%28%28a%2Bb%29%2F2%29%2Acos%28%28a-b%29%2F2%29 ) = 

                         = 2*sin(30°)*cos(17°) = 2%2A%281%2F2%29%2Acos%2817%5Eo%29 = cos(17°)


     THEREFORE, the sum of four terms in the numerator is  3*cos(17°).




(3)  Then it becomes OBVIOUS that the entire fraction is  equal to 3.     ANSWER


Solved.