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Magnitude of proper motion

Proper motion is the astronomical measure of the observed changes in the apparent places of stars or other celestial objects in the sky, as seen from the ... more

Worksheet 334

In a video game design, a map shows the location of other characters relative to the player, who is situated at the origin, and the direction they are facing. A character currently shows on the map at coordinates (-3, 5). If the player rotates counterclockwise by 20 degrees, then the objects in the map will correspondingly rotate 20 degrees clockwise. Find the new coordinates of the character.

To rotate the position of the character, we can imagine it as a point on a circle, and we will change the angle of the point by 20 degrees. To do so, we first need to find the radius of this circle and the original angle.

Drawing a right triangle inside the circle, we can find the radius using the Pythagorean Theorem:

Pythagorean theorem (right triangle)

To find the angle, we need to decide first if we are going to find the acute angle of the triangle, the reference angle, or if we are going to find the angle measured in standard position. While either approach will work, in this case we will do the latter. By applying the cosine function and using our given information we get

Cosine function
Subtraction

While there are two angles that have this cosine value, the angle of 120.964 degrees is in the second quadrant as desired, so it is the angle we were looking for.

Rotating the point clockwise by 20 degrees, the angle of the point will decrease to 100.964 degrees. We can then evaluate the coordinates of the rotated point

For x axis:

Cosine function

For y axis:

Sine function

The coordinates of the character on the rotated map will be (-1.109, 5.725)

Reference : PreCalculus: An Investigation of Functions,Edition 1.4 © 2014 David Lippman and Melonie Rasmussen
http://www.opentextbookstore.com/precalc/
Creative Commons License : http://creativecommons.org/licenses/by-sa/3.0/us/

Magnitude of proper motion (μα*)

Proper motion is the astronomical measure of the observed changes in the apparent places of stars or other celestial objects in the sky, as seen from the ... more

Tractive Force

As used in mechanical engineering, the term tractive force can either refer to the total traction a vehicle exerts on a surface, or the amount of the total ... more

Earth Similarity Index

The Earth Similarity Index, ESI or “easy scale” is a measure of how physically similar a planetary-mass object is to ... more

Boiling point elevation (ebullioscopy)

Colligative properties are properties of solutions that depend upon the ratio of the number of solute particles to the number of solvent molecules in a ... more

Dolbear's Law - in degrees Celsius

Dolbear’s law states the relationship between the air temperature and the rate at which crickets chirp. It was formulated by Amos ... more

Dolbear's Law - in degrees Fahrenheit

Dolbear’s law states the relationship between the air temperature and the rate at which crickets chirp. It was formulated by Amos ... more

Leffler formula (estimation of Ideal body weight for children 0–10 years of age)

The term body weight is used colloquially and in the biological and medical sciences to refer to a person’s mass or weight. Body weight is one way of ... more

Simplified von Mises equation - Pure shear

RESTRICTIONS : σ₁ = σ₂ = σ₃ = 0, σ₃₁ = σ₂₃ = 0

The von Mises yield criterion suggests that the yielding of materials ... more

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