# Point rotating in a coordinate system

The point P rotates with angle α to point P'. the coordinates of the old P are x1 and x2 and for P': x'1 and x'2.
Prove that:
$$x'_1=x_1\cos\alpha+x_2\sin\alpha$$
$$x'_2=x_2\cos\alpha-x_1\cos\alpha$$

I drew on the left the problem and on the right my attempt. the line OA, which is made of ##x_1\cos\alpha## plus ##x_2\sin\alpha## which is the blue line is indeed x'1 but i don't see the congruent triangles.

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