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Half-Life

Half-life is the amount of time required for the amount of something to fall to half its initial value. The term is very commonly used in nuclear physics ... more

Shear modulus (related to Young's modulus and Poisson's ratio)

The shear modulus is one of several quantities for measuring the stiffness of materials and describes the material’s response to shear stress (like ... more

Cross-multiplication - Rule of Three

In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can ... more

Mechanical Efficiency (pumps)

Mechanical components – as transmission gear and bearings – generates a mechanical loss that reduces the power transferred from the motor shaft ... more

Volume of a cube

A cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex.
The cube is also a square ... more

Malus' law (polarized light)

A polarizer or polariser is an optical filter that passes light of a specific polarization and blocks waves of other polarizations.
When a perfect ... more

Rayleigh Scattering - Intensity of Light

Rayleigh scattering (pronounced /ˈreɪli/ RAY-lee), named after the British physicist Lord Rayleigh (John William Strutt), is the (dominantly) elastic ... more

Dynamic Pressure - Compressible flow

In incompressible fluid dynamics dynamic pressure (indicated with q, or Q, and sometimes called velocity pressure) is the quantity defined as ... 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/

Transistor regulator ( Rv providing a bias current)

In the simplest case a common collector transistor (emitter follower) is used with the base of the regulating transistor connected directly to the voltage ... more

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