Section 4.1 Angles and Their Measures Trigonometry- 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 IIIIV 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

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 demoradian demo One full revolution = radians. One half revolution = radians. One quarter revolution = radians Complementary Angles: sum = Supplementary Angles: sum = Coterminal Angles: add/subtract

CONVERTING DEGREES TO RADIANS AND RADIANS TO DEGREES DEGREES RADIANS Multiply by RADIANS DEGREES Multiply by

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. 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.

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:

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