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Conic section (polar system and one focus on the pole and the other somewhere on the 0° ray )

conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. A conic ... more

Hypotrochoid (parametric equation Y- coordinate)

A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is ... more

Ball Screw - Leading Angle

A ball screw is a mechanical linear actuator that translates rotational motion to linear motion with little friction. A threaded shaft provides a helical ... 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/

Center of mass - Barycentric coordinates

In physics, the center of mass of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to ... more

Prandtl–Meyer expansion fan - Maximum turn angle

A supersonic expansion fan, technically known as Prandtl–Meyer expansion fan, is a centred expansion process that occurs when a supersonic flow turns ... more

Prandtl–Meyer expansion fan - Prandtl–Meyer function

A supersonic expansion fan, technically known as Prandtl–Meyer expansion fan, is a centred expansion process that occurs when a supersonic flow turns ... more

Möbius strip (y- coordinate )

The Möbius strip or Möbius band, is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being ... more

Möbius strip (z- coordinate )

The Möbius strip or Möbius band, is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being ... more

Möbius strip (x- coordinate )

The Möbius strip or Möbius band, is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being ... more

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