Section 4.1

Angles and Radian Measure

The Vocabulary of Angles • • • • An angle is formed by two rays that have a common endpoint.

One ray is called the initial side and the other is called the terminal side.

The common endpoint is called a vertex.

Angles are often named with lowercase Greek letters.

Some familiar vocabulary • • • • • • Angles are measured in degrees.

A complete rotation measures 360 degrees.

An acute angle measures more than 0 but less than 90 degrees.

An obtuse angle measures more than 90, but less than 180 degrees.

A right angle measures 90 degrees.

A straight angle measures 180 degrees.

Radians • • • • Angles are also measured in radians.

To define a radian, we must first define a central angle (a central angle is an angle who vertex is the center of a circle)… …and an intercepted arc (an intercepted arc is the portion of the circle between the two rays of a central angle).

One radian is the measure of a central angle whose intercepted arc is the same length as the radius of the circle.

The relationship between radians and degrees 360   2 

radians

Other equivalences follow from dividing the circle into “reasonable” portions.

Conversions • To convert from degrees to radians multiply by  180  • To convert from radians to degrees multiply by 180  

More vocabulary • An angle is in standard position if 1. Its vertex is at the origin • • 2. Its initial side lies along the positive x-axis Counterclockwise rotation from the initial side to the terminal side means the angle is positive.

Clockwise rotation from the initial side to the terminal side means the angle is negative.

Still more vocabulary • • When the terminal side of an angle lies in a quadrant, we say that the angle lies in that quadrant.

When the terminal side of an angle lies on the x-axis or y-axis, we say that the angle is quadrantal.

Draw the following and give the quadrant or axis 270   60  5  6 radians   radians

Coterminal Angles • • Two angles with the same initial and terminal sides but possibly different rotations are called coterminal angles.

Adding or subtracting a multiple of 360 degrees or 2”pi” radians will result in a coterminal angle.

Examples • Find a coterminal angle between 0 and 360 degrees or 0 and 2”pi” radians for each of the following: 570   70  17  5  11  3

Arc Length • Consider a circle of radius r with a central angle of measure “theta” (in radians). The measure s of the intercepted arc is given by

s



r



Examples

1. Find s if r = 4 cm and “theta” = 2 radians 2. Find r if s = 10 in and “theta” = 3 radians 3. Find “theta” if s = 7.5 m and r = 5 m 4. Find s if r = 8 ft and “theta” = 120 degrees

Converting from DMS to Decimal Degrees (and vice versa) • • • • • “DMS” stands for Degrees-Minutes-Seconds.

The conversions are based on subdivisions of a whole degree: 1 degree = 60 minutes 1 minute = 60 seconds It follows that 1 degree = 3600 seconds

From DMS to DD

D



M

'

S

" 

D



M

60 

S

3600 29

From DD to DMS 1. Multiply the decimal part of the degrees (to the right of the decimal point) by 60. These are your minutes.

2. Now multiply the decimal part of your minutes (to the right of the decimal point) by 60. These are your seconds.

Examples • 35 degrees 23 minutes 57 seconds to DD (round to four decimal places) • 97.55 degrees to DMS