Final answer:
To find sin x and cos x, we use the lengths of the sides of the triangle formed by the rectangle ABCD. The lengths are AB = CD = 6 cm and BC = 2 cm. We find that sin x = √10/3 and cos x = 1/3. The length of line OZ can be found using the Pythagorean theorem as OC = 2√19 cm.
Step-by-step explanation:
In order to find the values of sin x and cos x, we first need to find the lengths of the sides of the triangle. We have BC = 2 cm and CD = 6 cm. Since ABCD is a rectangle, AB = CD = 6 cm. Using the Pythagorean theorem, we can find the length of the other side AC: AC^2 = AB^2 + BC^2 = 6^2 + 2^2 = 40. Taking the square root of both sides, we find that AC = √40 = 2√10 cm.
In the right triangle ACO, where OC is the hypotenuse and AC and AO are the other two sides, we can use the sine and cosine ratios to find sin x and cos x:
sin x = AC/OA = 2√10/6 = √10/3
cos x = AO/OA = BC/OA = 2/6 = 1/3
To find the line OZ, we need to determine the length of OC. Since OC is the hypotenuse of triangle ACO, which is a right triangle, we can use the Pythagorean theorem:
OC^2 = AC^2 + AO^2 = 40 + 36 = 76. Taking the square root of both sides, we find that OC = √76 = 2√19 cm.