Find the Angle Whose Sine is the Product of the Sine and Cosine of Two Other Angles

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This Demonstration shows a geometric solution of the trigonometric equation , where is unknown.


Draw a right-angle triangle with the hypotenuse and . Then . Let be the circumcircle of .

Transform the equation to and interpret it as the law of sines for a triangle with of length 1 opposite the angle and side of length opposite the unknown angle .

To find the point , draw a circle with inscribed angle over the chord . Let be the center of . Construct the point on so that . By the law of sines, the angle at is a solution for .

If is the intersection of and , then . You can choose to show the point .


Contributed by: Izidor Hafner (July 2017)
Open content licensed under CC BY-NC-SA



The idea for this Demonstration comes from [1].


[1] J. Plemelj, Iz mojega življenja in dela (From My Life and Work), Obzornik mat. fiz., 39, 1992 pp. 188–192.

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