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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

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
Creative Commons License : http://creativecommons.org/licenses/by-sa/3.0/us/

Hick's Law

Hick’s law, or the Hick–Hyman law, named after British and American psychologists William Edmund Hick and Ray Hyman, describes the time it takes for ... more

Optical depth

In physics, optical depth or optical thickness, is the natural logarithm of the ratio of incident to transmitted radiant power through a material, and ... more

Radio luminosity

In astronomy, luminosity is the total amount of energy emitted by a star, galaxy, or other astronomical object per unit time. It is related to the ... more

Worksheet 306

Calculate the force the biceps muscle must exert to hold the forearm and its load as shown in the figure below, and compare this force with the weight of the forearm plus its load. You may take the data in the figure to be accurate to three significant figures.

(a) The figure shows the forearm of a person holding a book. The biceps exert a force FB to support the weight of the forearm and the book. The triceps are assumed to be relaxed. (b) Here, you can view an approximately equivalent mechanical system with the pivot at the elbow joint


There are four forces acting on the forearm and its load (the system of interest). The magnitude of the force of the biceps is FB, that of the elbow joint is FE, that of the weights of the forearm is wa , and its load is wb. Two of these are unknown FB, so that the first condition for equilibrium cannot by itself yield FB . But if we use the second condition and choose the pivot to be at the elbow, then the torque due to FE is zero, and the only unknown becomes FB .


The torques created by the weights are clockwise relative to the pivot, while the torque created by the biceps is counterclockwise; thus, the second condition for equilibrium (net τ = 0) becomes

Force (Newton's second law)
Force (Newton's second law)

Note that sin θ = 1 for all forces, since θ = 90º for all forces. This equation can easily be solved for FB in terms of known quantities,yielding. Entering the known values gives

Mechanical equilibrium - 3=3 Torque example

which yields


Now, the combined weight of the arm and its load is known, so that the ratio of the force exerted by the biceps to the total weight is



This means that the biceps muscle is exerting a force 7.38 times the weight supported.

Reference : OpenStax College,College Physics. OpenStax College. 21 June 2012.
Creative Commons License : http://creativecommons.org/licenses/by/3.0/

Ratio between two field quantities expressed in decibels

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Richter magnitude scale

assigns a magnitude number to quantify the energy released by an earthquake. The Richter scale is a base-10 logarithmic scale, which defines magnitude as ... more

Logarithm of the odds ratio

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Dividend cover ratio

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Sersic profile

The Sérsic profile (or Sérsic model or Sérsic’s law) is a mathematical function that describes how the intensity I of a galaxy varies with distance ... more

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