13.2 Define General Angles and Use Radian Measure

Initial side and terminal side

An angle can be formed by fixing one ray, called the initial side, and rotating the other ray, called the terminal side, about the vertex.

Standard position

The position of an angle whose vertex is at the origin and its initial side lies on the positive x-axis

Coterminal

Angles in standard position whose terminal sides coincide

Radian

In a circle with radius r centered at the origin, one radian is the measure of an angle in standard position whose terminal side intercepts an arc of length r.

Sector

A region of a circle that is bounded by two radii and an arc of the circle

Central angle

The angle formed by two radii of a circle

ANGLES IN STANDARD POSITION

In a coordinate plane, an angle can be formed by fixing one ray, called the _initial_ side, and rotating the other ray, called the _terminal_ side, about the _vertex_.

An angle is in standard position if its vertex is at _the origin_ and its initial side lies on the positive _x-axis_ .

Example 1

Draw angles in standard position

Draw an angle with the given measure in standard position.

a. 405°

b. -65°

a. Because 405° is _45°_ more than 360°, the terminal side makes one whole revolution _counterclockwise_ plus _45°_ more.

b. Because -65° is negative, the terminal side is _65° clockwise_ from the positive
x-axis.

Example 2

Find coterminal angles

Find one positive angle and one negative angle that are coterminal with 210°.

There are many such angles, depending on what multiple of 360° is added or subtracted.

210° + 360° = _570°_ 210° - 360° = _-150°_

CONVERTING BETWEEN DEGREES AND RADIANS

Degrees to radians

Multiply degree measure by	p radians	.
	180°

Radians to Degrees

Multiply radian measure by	180°	.
	p radians

Example 3

Convert between degrees and radians

Convert (a) 315° to radians and (b) radians to degrees.

a. 315° = 315°	p radians	=	7 p	radians
	180°		4

b.	radians	p radians	=	_30^o_
		180°

ARC LENGTH AND AREA OF A SECTOR

The arc length s and area A of a sector with radius r and central angle q (measured in radians) are as follows.

Arc length: s = rq

Area: A = r²q

Example 4

Solve a multi-step problem

Find the arc length and area of a sector with a radius of 15 inches and a central angle of 60°.

Solution

1. Convert the measure of the central angle to radians.

60^o = 60^o	p radians	=	_____ radians
	180°

2. Find the arc length and the area of the sector. Arc length: s = rq

=__15__

= __5p__

» __15.71__ inches

Area: A = r²q

= (_15_)²

= _37.5 p_

» _117.81_ square inches.