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Radian Measure and the Circular Functions

This page is meant to serve as a quick overview of radian measurement and the circular functions.

**Radian Measure**

Radian measure of any angle is the length of the arc intercepted on a circle with a radius of 1 by an angle in the standard position on a coordinate plane.

**Degree / Radian Relationship**

180° = π radians

**Degree / Radian Conversions**

1. To convert a radian measure to degrees, multiply a radian measure by 180°/π and simplify.

2. To convert a degrees measure to radians, multiply a radian measure by π/180° and simplify.

**Arc Length**

The arc length, *s*, is the measure of the distance along the curved line making up the arc. The length of the arc, *s*, intercepted on a circle by a central angle of Θ, can be determined by *s*=*rΘ*

**Area of Sector**

The arc length, *s*, is the measure of the distance along the curved line making up the arc. The length of the arc, *s*, intercepted on a circle by a central angle of Θ, can be determined by *A*=*(1/2)(r²Θ)*

**The Unit Circle**

**Cicular Functions**

For the six circular functions, start with the point (1,0) and \lay off an arc length of s along the circle. If s is counterclockwise, it is positive, where if its clockwise, it is negative. Let the entpoint of the arc be (x,y).

sin s = y | cos s = x | tan s = y / x |

csc s = 1 / y | sec s = 1 / x | cot s = x / y |

**Domains of the Circular Functions**

So long that *n* is any integer, the somains of the circular functions are as follows

Sine and Cosine Functions | (-∞,∞) |

Tangent and Secant Functions | {s|s ≠ π/2 + nπ} |

Cotangent and Cosecant Functions | {s|s ≠ nπ} |

**Angular Velocity**

ω = Θ / *t*

(ω in radians per unit of time with Θ in radians)

**Linear Velocity**

*v* = *s* / *t*

*v* = *rΘ* / *t*

*v* = * rω*