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Externally Tangent Circles

Two circles of non-equal radius, both in the same plane, are said to be tangent to each other if they meet at only one point.
Two circles are ... more

Internally Tangent Circles

Two circles of non-equal radius, both in the same plane, are said to be tangent to each other if they meet at only one point.
Two circles are ... more

Basic reproduction number for a population with an exponential age distribution

Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic and help inform public health interventions. ... more

Resonant frequency of string

Acoustic resonance is the tendency of an acoustic system to absorb more energy when it is forced or driven at a frequency that matches one of its own ... more

Angular resolution

Angular resolution or spatial resolution describes the ability of any image-forming device such as an optical or radio telescope, a microscope, a camera, ... more

Speed of sound in sea water (UNESCO equation)

The speed of sound is the distance travelled per unit of time by a sound wave propagating through an elastic medium. The speed of sound in seawater depends ... more

R-value (insulation) of a multi-layered installation

Formula first contributed by:
zfyl

The R-value is a measure of thermal resistance, or ability of heat to transfer from hot ... 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/

Time dilation due to fthe time interval between two co-local events

Time dilation is an actual difference of elapsed time between two events as measured by observers either moving relative to each other or differently ... more

Sherwood Number for a single sphere

The Sherwood number (Sh) is a dimensionless number used in mass-transfer operation. It can be further defined as a function of the Reynolds and Schmidt ... more

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