A radian is the size of the angle subtended by an arc the same length as the radius of the circle.

Since the circumference of a circle is 2πr, there are 2π radians in 360°.

One radian is equal to ${360°}/{2π} = 57.2957795°$. Since π is irrational, the radian cannot be expressed exactly in degrees.

The length of an arc = $({θ}/{2π})(2πr) = rθ$

This formula assumes the angle is given in radians.

The area of a sector = $({θ}/{2π})(πr^2)={θr^2}/2$

where r the radius of the circle, and the sector subtends the central angle θ.

If $-2π ≤ x ≤ 2π$, there are a number of solutions to an equation such as: $sinx=1/{√2}$

The first quadrant solution is $π/4$ (45°). However, the sine of $π - π/4 = {3π}/4$, in the second quadrant, is also $1/{√2}$. But also sin${-5π}/4$ and sin${-7π}/4$ give solutions of $1/{√2}$.

If the domain is not limited to one cycle ($-2π ≤ x ≤ 2π$), then sin${9π}/4$, sin${11π}/4$, sin${17π}/4$, sin${19π}/4$, etc. are also solutions. And the periodic function could be extended in the negative direction as well: sin${-13π}/4$, sin${-15π}/4$, etc.

A system with periodic motion can be described by an equation. If the motion is simple harmonic or rotational, the equation can be a sinusoidal function of time, t, the starting position, h(0), and a periodic factor. An example is a Ferris Wheel:

$$H(t)=rsin({2π}/T(t-T/4))+(H(0) + r)$$where T is the period of one rotation, H(0) is the starting height, and r the radius.

$h(t)=60cos({2π}/{30}(t-15))+ 60$

At time $t=0$, the equation reduces to $h(t)=60cos(-π)+ 60 $

$= -60 + 60 = 0$: the starting position is 0.

At time $t=15$, the equation reduces to $h(t)=60cos(0)+ 60 $

$= 60 + 60 = 120$: the height at $t=15$ seconds is 120m. Since the maximum value of cos(x) is 1, this is the maximum height reached.

At time $t=30$, the equation reduces to $h(t)=60cos(π)+ 60 $

$= -60 + 60 = 0$: the height at $t=30$ seconds is once again 0m. The motion is periodic with a period of one cycle of 30 seconds.

Since a sine or cosine can take a value of -1, the zero point is established by $M$ = maximum height.

The angular speed of the motion is described by the argument of the cosine or sine: in our example $({2π}/{30}(t-15))$. In other words, a full cycle (2π radians) is made every $p$ seconds ($p$ = period).

The phase shift, $s$, establishes the starting time.

The general formula for position is:

$$P(t) = M⋅cos({2π}/{p}(t-s))+ M$$Content © Andrew Bone. All rights reserved. Created : April 1, 2015

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Physics is the science of the very small and the very large. Learn about Isaac Newton, who gave us the laws of motion and optics, and Albert Einstein, who explained the relativity of all things, as well as catch up on all the latest news about Physics, on ScienceLibrary.info.

1938 - 2011

Lynn Margulis was an American biologist who is considered one of the leading figures in the study of symbiosis in biological evolution.

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