# Circular and Periodic Functions

## The Unit Circle

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

### 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.

### 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.

### 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\$\$

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