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 © Renewable.Media. All rights reserved. Created : April 1, 2015

The most recent article is:

View this item in the topic:

and many more articles in the subject:

Science resources on ScienceLibrary.info. Games, puzzles, enigmas, internet resources, science fiction and fact, the weird and the wonderful things about the natural world. Have fun while learning Science with ScienceLibrary.info.

c. 1445 - 1517

Luca Pacioli was an Italian mathematician and teacher, who published books in the Italian vernacular to popularise mathematics.

Website © renewable.media | Designed by: Andrew Bone