To determine the trajectory of the movement of a material point in the xoz plane that is subjected simultaneously to oscillations x (t) = a * sin π(t + 1) and y (t) = a * sin (πt + π / 2).

We are to show that the given parametric curve is a circle.

The trajectory of a circle with a radius r will satisfy the following relationship:

(with (x_c,y_c) being the center point)

We are given the x and y in a parametric form which can be further rewritten (using properties of sin/cos):

Squaring and adding both gives:

The last expression shows that the given parametric curve is a circle with the center (0,0) and radius A.

as we probably am aware that our atmosphere or air have various levels, each level has different gasses, and each gas has another porousness. at the point when the light originates from lower porousness to higher penetrability, it alters its course. with the goal that implies that when the light originates from the moon to the earth, it alters it's course a lot more occasions. it essentially implies that the state of the light piece of the moon is changed on the earth.

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joke:

h o h o hope you have a merry christmas!