A right-angled triangle $ABC$ is inscribed in a circle of radius $6$, and the measure of its angle $BAC$ is $60^{\circ}$. Calculate the area of this triangle.
Peter solved this problem as follows:
(1) First he drew this picture:
From the picture, he directly determined that $$ |AB|=12. $$
(2) Using the sine function, he found the length of the opposite side $BC$: $$ \begin{gather} \sin(60^{\circ})= \frac{|AB|}{|BC|} \cr |BC|=\frac{|AB|}{ \sin(60^{\circ}) }\cr |BC|=8\sqrt{3} \end{gather} $$
(3) Using the cosine function, he found the length of the adjacent side $AC$: $$ \begin{gather} \cos(60^{\circ})= \frac{|AC|}{|AB| } \cr |AC|=|AB| \cos(60^{\circ}) \cr |AC|=6 \end{gather} $$
(4) Finally, he calculated the area of the triangle: $$ P=\frac{1}{2}\cdot 6 \cdot 8 \sqrt{3}=24 \sqrt{3} $$ Is Peter’s result correct? Which answer is true?
Peter's result is not right. Petr made the example easier by drawing the side $AB$ so that it passes through the center of the circumscribed circle, which may not be the case in general. The error is in step (1).
Peter’s result is not correct. He made a mistake in step (2). It should have been: $$ \sin(60^{\circ})= \frac{|BC|}{|AB|} $$
Peter’s result is not correct. He made a mistake in step (3). It should have been: $$ \cos(60^{\circ})=\frac{|AB|}{|AC| } $$
Peter’s result is not correct. He made a mistake in step (4). He should have used the formula: $$ A=\frac12 |AB||AC| \cos(60^{\circ}) $$
$$ \begin{gather} \sin (60^{\circ})= \frac{|BC|}{|AB| } \cr |BC|=6\sqrt{3} \end{gather} $$ The correct result is: $$ A=\frac{1}{2}\cdot 6 \cdot 6 \sqrt{3}=18 \sqrt{3} $$
