Find the cosine of the angle between the ray OD and the positive semiaxis Ox, if D (2; -1).

If we mentally lower the perpendicular from point D to the positive semiaxis, then the coordinates of its intersection point (let it be A) with the semiaxis will be (2; 0).

In the formed triangle, side OD will be the hypotenuse. And the side OA is the leg that will be adjacent to the corner, whose cosine must be found.

Find out the length of the hypotenuse:

√ ((2 – 0) ^ 2 + (-1 – 0) ^ 2) = √5.

Find out the length of the leg:

√ ((2 – 0) ^ 2 + (0 – 0) ^ 2) = 2.

Then the cosine of the angle will be:
2: √5 = 2√5 / 5.

Answer: The cosine is 2√5 / 5.