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Sensitivity ( true positive rate)

For classification tasks, the terms true positives, true negatives, false positives, and false negatives compare the results of the classifier under test ... more

Gompertz–Makeham Law of Mortality

The Gompertz–Makeham law states that the human death rate is the sum of an age-independent component (the Makeham term, named after William Makeham) and an ... more

Generalized mean (power mean)

In mathematics, generalised means are a family of functions for aggregating sets of numbers, that include as special cases the arithmetic, geometric, and ... more

NPSH characterize the potential for cavitation. The suction head coefficient is a dimensionless measure of ... more

Weighted power mean

The weighted mean is similar to an arithmetic mean (the most common type of average), where instead of each of the data points contributing equally to the ... 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/

Acceleration of a simple harmonic motion

A simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement. The motion is sinusoidal in ... more

Velocity of a simple harmonic motion

A simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement. The motion is sinusoidal in ... more

True anomaly

In astronomy, Kepler’s laws of planetary motion are three scientific laws describing the motion of planets around the Sun.

1- The orbit of ... more

Frequency of a string under tension (nth harmonic)

A vibration in a string is a wave. Resonance causes a vibrating string to produce a sound with constant frequency, i.e. constant pitch. If the length or ... more

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