T maps , the vector from (0,0) to (1,0) into its projection on the line through the origin with angle . drawing a perpendicular from (1, 0) to that line gives a right triangle with hypotenuse 1 and angle . That is, the distance from (0, 0) to point of intersection is . If you now drop a perpendicular to the x-axis, you have a right triangle with angle and hypotenuse . The x and y coordinates of that point are the lengths of the "opposite" and "near" sides of that right triangle. That is, we have and . From that the (x,y) coordinates of the point are .
Now, do the same with the projection of (0, 1) onto that line.