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Ratio between two field quantities expressed in decibels

he decibel is a logarithmic unit used to express the ratio between two values of a physical quantity. The bel represents a ratio between two power ... more

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

Carnot cycle

Every single thermodynamic system exists in a particular state. When a system is taken through a series of different states and finally returned to its ... more

Sagnac Effect - TIme Difference

The Sagnac effect (also called Sagnac interference), named after French physicist Georges Sagnac, is a phenomenon encountered in interferometry that is ... more

Black hole surface gravity

The surface gravity, g, of an astronomical or other object is the gravitational acceleration experienced at its surface. The surface gravity may be thought ... more

Present value of an annuity for n payment periods

An annuity is a series of equal payments or receipts that occur at evenly spaced intervals. Leases and rental payments are examples. The payments or ... 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/

Wien's displacement law

Wien’s displacement law states that the black body radiation curve for different temperature peaks at a wavelength that is inversely proportional to ... more

Flow Rate (related to velocity)

Flow rate is defined to be the volume of fluid passing by some location through an area during a period of time.
The change in volume is the amount ... more

Rule of 70 ( estimating the time for money's buying power to halve)

Purchasing power (sometimes retroactively called adjusted for inflation) is the number of goods or services that can be purchased with a unit of currency. ... more

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