Science Library - free educational site

Circular and Periodic Functions

The Unit Circle

Mathematics trigonometry principles
Mathematics trigonometry principles
Mathematics trigonometry principles
Mathematics trigonometry principles

Radians

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.

Arcs

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

This formula assumes the angle is given in radians.

Sectors

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.

Graphing Circular Functions

Basic trig identities
Mathematics question

Solving equations with the unit circle

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.

Modelling with Sine and Cosine

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.

Mathematics question

Ferris Wheel Example

$h(t)=60cos({2π}/{30}(t-15))+ 60$

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.

Mathematics

General Periodic Motion

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.com. All rights reserved. Created : April 1, 2015

Latest Item on Science Library:

The most recent article is:

Trigonometry

View this item in the topic:

Vectors and Trigonometry

and many more articles in the subject:

Subject of the Week

Environment

Environmental Science is the most important of all sciences. As the world enters a phase of climate change, unprecedented biodiversity loss, pollution and human population growth, the management of our environment is vital for our futures. Learn about Environmental Science on ScienceLibrary.info.

Environmental Science

Great Scientists

Caroline Herschel

1750 - 1848

Caroline Herschel has traditionally been neglected by history, living in the shadow of her famous brother, William Herschel. But recent research has demonstrated that she made valuable contributions to science in her own right.

Caroline Herschel
Lugano English

Quote of the day...

Ancient or not, the Seulpierre family as you will find it today is not really that different to any other family, here or there. Which means it is in every way unique.

ZumGuy Internet Promotions

English language school