Section 4.1 Angles and Their MeasuresTrigonometry- measurement of angles

IMPORTANT VOCABULARY:

Angle- determined by rotating a ray about its endpoint.

Initial Side- the starting position of the ray.

Terminal Side- the position of the ray after rotation.

Standard Position Angle- any angle whose initial side lies on the positive

x-axis and vertex is at the origin when placed on the coordinate plane.

standard position angle

Positive Angle- generated by counterclockwise rotation.

Negative Angle- generated by clockwise rotation.

Coterminal Angles- angles which have the same initial and terminal side.

Initial side

Terminal side

Positive angle

Initial side

Terminal side

Negative angle

QUADRANTS

III

III IV

The quadrant an angle lies in is determined by the terminal side. An angle in standard position lies in the quadrant in which its terminal side lies in. Any angle which terminates on either the x or y axis is called a quadrant angle and does not lie in any quadrant.

Angles are labeled using Greek letters (alpha), (beta), and (theta), as well as uppercase letters A, B, and C.

Standard position

Acute Angle: measures between

Obtuse Angle: measures between

Right Angle: measures

Straight Angle: measures

To find coterminal angles in degrees, you add/subtract

Complementary Angles: two positive angles whose sum is

Supplementary Angles: two positive angles whose sum is

0 and 90

90 and 180

90

180

360

90

180

Another way to measure angles is in radians.

Radian: the measure of a central angle that intercepts an arc s equal in length to the radius r of the circle. radian demo

One full revolution = radians.

One half revolution = radians.

One quarter revolution = radians

Complementary Angles: sum =

Supplementary Angles: sum =

Coterminal Angles: add/subtract

360 2

180

902

2

2

CONVERTING DEGREES TO RADIANS AND RADIANS TO DEGREES

DEGREES RADIANS

Multiply by

RADIANS DEGREES

Multiply by

180

180

The length of the arc (s) of a circle is equal to the radian measure of the

angle ,in radians, times the radius of the circle.

s r

REMEMBER…..this formula speaks radian measures!!!!

If you are given an angle in degrees you must convert to radians before you can plug into this formula!!!

With calculators it is convenient to use decimal degrees to denote fractional parts of degrees. Historically, however, fractional parts of degrees were expressed in minutes and seconds, using the prime (׳) and double prime (״) notations.

1 minute =

1 second =

Consequently, an angle of 64 degrees, 32 minutes, and 47 seconds was

represented by

Calculators have special keys for converting DMS to Decimal Degrees and vice versa.

1 160

1 13600

64 34 47

The keys used to convert DMS to decimal degrees and vice versa are under the angle menu on the calculator. Before we begin make sure your calc is in the degree mode.

To get to the angle menu hit .

Here you will see that 1: is the degree symbol and 2: is the minute symbol.The seconds symbol is actually under the alpha mode and you get there by hitting .

To convert to decimal degrees you would do the following in your calculator:

64 34 47

To convert decimal degrees to DMS do the following in your calculator:

Convert to DMS.

27.345