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Pythagorean triangle ( "b" side)

A Pythagorean triangle is right angled and Heronian. Its three integer sides are known as a Pythagorean triple or Pythagorean triplet or Pythagorean triad. ... more

Pythagorean triangle ( hypotenuse)

A Pythagorean triangle is right angled and Heronian. Its three integer sides are known as a Pythagorean triple or Pythagorean triplet or Pythagorean triad. ... more

Right Triangle (sides)

A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree ... more

Triangle Wave

A triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.
Like a ... more

Sawtooth wave

The sawtooth wave (or saw wave) is a kind of non-sinusoidal waveform. The convention is that a sawtooth wave ramps upward and then sharply drops. However, ... more

Gas in a box (momentum)

The particle in a box model describes a particle free to move in a small space surrounded by impenetrable barriers. the results of the quantum particle in ... more

Rhodonea curve

In mathematics, a rose or rhodonea curve is a sinusoid plotted in polar coordinates. the polar coordinate system is a two-dimensional coordinate system in ... more

Triple-angle's cosine (related to the cosine of the single angle)

rigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, ... more

Triple-angle's sine (related to the sine of the single angle)

Trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. Geometrically, ... more

Law of cosines

The law of cosines relates the cosine of an angle to the opposite side of an arbitrary triangle and the length of the triangle’s sides.
The law ... more

Pythagorean theorem (right triangle)

In mathematics, the Pythagorean theorem, also known as Pythagoras’ theorem, is a fundamental relation in Euclidean geometry among the three sides of ... more

Triangle wave (in trigonometric terms)

A triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function. Like a square ... more

Beta Function

In mathematics, the beta function, also called the Euler integral of the first kind, is a special function.The beta function was studied by Euler and ... more

Logarithm of the odds ratio

The logit function is the inverse of the sigmoidal “logistic” function or logistic transform used in mathematics, especially in statistics. ... 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

Morley's trisector theorem (area)

Morley’s trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, ... more

Gamma Function

In mathematics, the gamma function (represented by the capital Greek letter Γ) is an extension of the factorial function, with its argument shifted down by ... more

Morley's trisector theorem

Morley’s trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral ... more

Pythagorean theorem (arbitrary triangle - acute angle)

Generalization of the Pythagorean theorem for the side opposite of the acute angle of an arbitrary triangle

... more

Pythagorean theorem (arbitrary triangle - obtuse angle)

Generalization of the Pythagorean theorem for the side opposite of the obtuse angle of an arbitrary triangle

... 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/

Factorial

The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. The value of 0! is 1, according ... more

Miller indices calculator ( planar spacing distance in bcc system)

Miller indices form a notation system in crystallography for planes in crystal (Bravais) lattices.
In particular, a family of lattice planes is ... more

Miller indices calculator ( planar spacing distance in fcc system)

Miller indices form a notation system in crystallography for planes in crystal (Bravais) lattices.
In particular, a family of lattice planes is ... more

Miller indices calculator (Case of cubic structures)

Miller indices form a notation system in crystallography for planes in crystal (Bravais) lattices.
In particular, a family of lattice planes is ... more

Rydberg formula for any hydrogen-like element

The Rydberg formula is used in atomic physics to describe the wavelengths of spectral lines of many chemical elements.Rydberg worked on a formula ... more

Bragg's Law

In physics, Bragg’s law, or Wulff–Bragg’s condition, a special case of Laue diffraction, gives the angles for coherent and incoherent ... more

IT Grade

IT Grade refers to the International Tolerance Grade of an industrial process defined in ISO 286. This grade identifies what ... more

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