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Subtraction - 10 objects/subtrahend version

Subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then ... more

Mean Time Between Failures - MTBF

Mean time between failures (MTBF) is the predicted elapsed time between inherent failures of a system during operation. ... more

Proper motion (right ascension)

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

Geometric mean of two numbers

The geometric mean is defined as the square root of the product of the numbers. The geometric mean only applies either to positive numbers or both negative ... more

Capital asset pricing model

In finance, the capital asset pricing model is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be ... more

Cost of equity

The cost of capital is a term used in the field of financial investment to refer to the cost of a company’s funds (both debt and equity). Equity is ... more

Population growth rate - Logistic equation

In biology or human geography, population growth is the increase in the number of individuals in a population.

The “population growth ... 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

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
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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

AC Power

In alternating current circuits, energy storage elements such as inductance and capacitance may result in periodic reversals of the direction of energy ... more

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