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Geometry Level pending

The two sides of a triangle measures 18 and 14 units with an angle of 10.8 4 0 10.84^0 . Evaluate sin π a \frac{π}{a} - sin 2 π a \frac{2π}{a} - sin 3 π a \frac{3π}{a} - . . . - sin 8 π a \frac{8π}{a} + sin 9 π a \frac{9π}{a} , if a is equal to the third side of the triangle.


The answer is 0.

John Paolo Bayaborda by John Paolo Bayaborda , 24, Philippines

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1 solution

using the formula, a 2 a^2 = b 2 b^2 + c 2 c^2 - 2bc Cos A, we can solve the value of the third side.

a 2 a^2 = 1 8 2 18^2 + 1 4 2 14^2 - 2(18)(14)(Cos 10.8 4 0 10.84^0 )

therefore, a = 5

by substitution, sin π 5 \frac{π}{5} - sin 2 π 5 \frac{2π}{5} - sin 3 π 5 \frac{3π}{5} - . . . - sin 8 π 5 \frac{8π}{5} + sin 9 π 5 \frac{9π}{5}

so, the answer is 0.

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