Lesson 5-1: Angles and Degree Measure Pre- Calculus Lesson 5-1: Angles and Degree Measure   Vertex:   Initial Side:   Terminal Side   Standard Position: Degree: Minutes: Seconds: The endpoint of two rays that form an angle. Terminal Side The side of the angle that is fixed. Initial Side The side of the angle that will rotate or move. Vertex An angle with its vertex at the origin and its initial side along the positive x-axis. Common measure used to measure an angle. Degree is based on 60 rather than 10 as we use in decimals. Therefore 1o is equal to 1 60 of the measure of equilateral triangle. The degree is subdivided into 60 equal parts called minutes. The minute is divided into 60 equal parts known as seconds.

If rotation is clockwise it is a negative angle. Example 1: Geographic locations are typically expressed in terms of latitude and longitude.   a. Las Vegas, Nevada, is located at about 36.175° north latitude. Change to 36.175° to degrees, minutes, and seconds. b. Las Vegas is also located at 115° 8 11 west longitude. Write 115 ° 8 11 as a decimal rounded to the nearest thousandth. 36.175 = 36 o + (0.175 . 60)’ = 36o + 10.5’ 36o 10’ 30” = 36o + 10’ + (0.5 . 60)” = 36o + 10’ + 30” 115o + (8 . 1 60 )’ + (11 . 1 3600 )” About 115.1363889… 115o + 0. 133333…. + 0.003055555… About 115.136 One Rotation of a Circle =   Clockwise = Counterclockwise = 360o If rotation is clockwise it is a negative angle. If rotation is counterclockwise it is a positive angle.

Two angles in standard position, if they have the same terminal side. Example 2: Give the angle measure represented by each rotation.   a. 3.75 rotations clockwise b. 4.2 rotations counterclockwise 3.75 . (-360) 4.2 . 360 -1350o 1512o Coterminal angles:   If ∝ is the degree measure of an angle, then all angles measuring ∝ + 360k, where k is an integer, are coterminal with ∝. Example 3: Identify all angles that are coterminal with each angle. Then find one positive angle and on negative angle that are coterminal with the angle. a. 42° b. 128° Two angles in standard position, if they have the same terminal side. All angles having a measure of 42o + 360ko, where k is an integer. All angles having a measure of 128o + 360k, where k is an integer. Positive Angle is 42o + 360(2) = 42 + 720 = 762o Positive Angle is 128o + 360(3) = 128 + 1080 = 1208o Negative Angle is 42o + 360(-2) = 42 + (-720) = -678o Negative Angle is 128 + 360(-3) = 128 + -1080 = -952o

Take 445 360 to determine how many complete rotations there are. 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.   a. 445° b. -2408° Take 445 360 to determine how many complete rotations there are. Take −2408 360 = -6.68888888… 1.236111… -.68888888 ∙360= About -248 Now determine the remaining degrees of .236111 by .236111 ∙ 360 = About 85o, Quadrant I Coterminal angle needs to be positive: 360 – 248 = 112o , Quadrant II Reference Angle: Reference Angle Rule For any angle ∝, 0o < ∝ < 3600, its reference angle ∝ ′ is defined by: a. ∝ , when the terminal side is in Quadrant I b. 180 - ∝ , when the terminal side is in Quadrant II c. ∝ - 180 , when the terminal side is in Quadrant III d. 360 - ∝ , when the terminal side is in Quadrant IV The acute angle formed by the terminal side of the given angle and the x-axis.

Example 5: Find the measure of the reference angle for each angle.   b. -305° Since 240o is between 180 and 270 the terminal side of the angle lies in Quadrant III. A coterminal angle of -305 is 360 – 305 = 55 Since 55o is between 90 and 0, the terminal side of the angle lies in Quadrant I. 240 – 180= 60o The reference angle is 55o.