Question 1120265
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(1) If x is positive and less than pi/2, then x and (pi/2-x) are complementary acute angles; the sine of one is equal to the cosine of the other.<br>
Specifically, cos(pi/2-x) = sin(x); the equation then says sin(pi/3) = sin(x); that makes the answer x = pi/3.<br>
(2) Using the formula for the cosine of the difference of two angles gives the same result:<br>
cos(pi/2-x) = cos(pi/2)*cos(x)+sin(pi/2)*sin(x) = 0+sin(x) = sin(x)<br>
And again the equation says sin(pi/3) = sin(x), so the answer is  = pi/3.<br>
(3) However, x=pi/3 is only what might be called the "principal" answer; there are infinitely many more.<br>
sin(pi/3) = sqrt(3)/2) = cos(pi/6) --> pi/2-x = pi/6 --> x = pi/3
sin(pi/3) = sqrt(3)/2) = cos(11pi/6) --> pi/2-x = 11pi/6 --> x = -4pi/3
sin(pi/3) = sqrt(3)/2) = cos(13pi/6) --> pi/2-x = 13pi/6 --> x = -5pi/3
sin(pi/3) = sqrt(3)/2) = cos(-pi/6) --> pi/2-x = -pi/6 --> x = 2pi/3
sin(pi/3) = sqrt(3)/2) = cos(-11pi/6) --> pi/2-x = pi/6 --> x = 7pi/3
sin(pi/3) = sqrt(3)/2) = .....  etc., etc., ....<br>