Chapter 5-1 Angles and Degree Measure

Degree is most commonly used to measure angles. Ancient Babylonian culture based their numeration on 60 rather than 10 and they assigned the measure of each angle of an equilateral triangle to be 60 degrees. Therefore, one sixtieth of the measure of the angle of an equilateral triangle is equal to one unit or degree. The degrees are subdivided into 60 equal parts called minutes and the minute is divided into 60 equal parts called seconds. Latitude and longitude can either be expressed in degrees as a decimal, or in degrees, minutes, and seconds.

Vertex: a fixed endpoint from which an angle is generated by rotating on e of two rays Initial side: One of the fixed rays that forms the angle Terminal side: the ray that rotates to form the terminal side If the rotation is counterclockwise, the angle formed is positive. If the rotation is clockwise, the angle formed is negative. Standard position: an angle with its vertex at the origin and the initial side along the positive x axis 120° Terminal Side Initial side Angle vertex Initial Side Terminal side -120 degrees

Example 1 Navigation At a certain time during a flight, an airplane is on an east-west line between two signal towers. If the bearing from the plane to tower A is 329.125 degrees, change it to degrees, minutes, and seconds. degrees 329 minutes .125* 60 = 7.5 use decimal number for seconds seconds .5 * 60 = 30 b. If the bearing from the plane to tower B is 35 degrees, 12 minutes, 7 seconds, write the bearing as a decimal rounded to the nearest thousandth. 35 whole number 12 * 1/60 = .2 7 * 1/3600 = .0019 Nearest thousandth Add 35 + .2 + .02 = 35.202 329 ° 7 ‘ 30 “

Quadrantal Angles-when the terminal side of an angle is in standard position and coincides with one of the axes. A full rotation around a circle is 360 degrees. A measure of more than 360 degrees represent multiple rotations. A rotation clockwise is – and counterclockwise is +.

Example 2 Give the angle measure represented by each rotation. 9.5 rotations clockwise 9.5 *- 360° = -3420° 6.75 rotations counterclockwise 6375 * 360° = 2430°

Coterminal angles: two angles in standard position that have the same terminal side. Every angle has infinitely many coterminal angles because every angle has many differing degree measures by multiples of 360 degrees. Formula: if x is the degree measure of an angle, then all angles measuring x + 360°k, where k is an integer, are coterminal with x.

Example 3 Identify all angles that are coterminal with each angle. Then find one positive angle and one negative angle that are coterminal with the angle. 86° 86 + 360(1)° = 446° 86 + 360(-1)° = -274° 294° 294 + 360(1)° = 654° 294 + 360 (-1)° = -66° Example 4 If each angle is in standard position, determine a coterminal angle that is between 0° and 360°. State the quadrant in which the terminal side lies. 595° 595/360 = 1.65277778 x + 360 (1)° = 595° .65277778 * 360° = 235° - 360 -360 x = 235° -777°

If x is a nonquadrantal angle in standard position, its reference angle is defined as the acute angle formed by the terminal side of the given angle and the x-axis. Reference Angle Rule Terminal side in Quad II 180° - x Terminal side in Quad I x Terminal side in Quad III x - 180° Terminal side in Quad IV 360° - x

Example 5 Find the measure of the reference angle for each angle. 312° Quadrant III 360° - 312° = 48° b. -195° Quad III Make it a positive angle first 360° - 195° = 165° 180° - 165° = 15°