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Pre- Calculus Lesson 5-1: Angles and Degree Measure Vertex: The endpoint of two rays that form an angle. Terminal Side Initial Side: Terminal Side The side of the angle that is fixed. The side of the angle that will rotate or move. Standard Position: Vertex Initial Side 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 Degree: based on 60 rather than 10 as we use in decimals. Therefore 1 1o is equal to of the measure of equilateral triangle. 60 Minutes: The degree is subdivided into 60 equal parts called minutes. Seconds: The minute is divided into 60 equal parts known as seconds. Example 1: a. Geographic locations are typically expressed in terms of latitude and longitude. Las Vegas, Nevada, is located at about 36.175° north latitude. Change to 36.175° to degrees, minutes, and seconds. 36.175 = 36 o + (0.175 . 60)’ = 36o + 10.5’ 36o 10’ 30” = 36o + 10’ + (0.5 . 60)” = 36o + 10’ + 30” b. Las Vegas is also located at 115° 8 11 west longitude. Write 115 ° 8 11 as a decimal rounded to the nearest thousandth. 1 1 115o + (8 . 60)’ + (11 . 3600 )” About 115.1363889… 115o + 0. 133333…. + 0.003055555… One Rotation of a Circle = Clockwise = About 115.136 360o If rotation is clockwise it is a negative angle. Counterclockwise = If rotation is counterclockwise it is a positive angle. Example 2: a. Give the angle measure represented by each rotation. 3.75 rotations clockwise b. 4.2 rotations counterclockwise 3.75 . (-360) 4.2 . 360 -1350o Coterminal angles: 1512o Two angles in standard position, if they have the same terminal side. If ∝ is the degree measure of an angle, then all angles measuring ∝ + 360k, where k is an integer, are coterminal with ∝. Example 3: a. Identify all angles that are coterminal with each angle. Then find one positive angle and on negative angle that are coterminal with the angle. 42° b. • All angles having a measure of where k is an integer. • Positive Angle is 42o + 360(2) = 42 + 720 = 762o 42o + 360ko, 128° • All angles having a measure of 128o + 360k, where k is an integer. • Positive Angle is 128o + 360(3) = 128 + 1080 = 1208o • Negative Angle is 42o + 360(-2) = 42 + (-720)• Negative Angle is 128 + 360(-3) = 128 + -1080 = -952o = -678o Example 4: a. 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. 445° b. 445 Take to determine how many complete 360 rotations there are. 1.236111… Now determine the remaining degrees of .236111 by .236111 ∙ 360 = About 85o, Quadrant I Reference Angle: -2408° Take −2408 360 = -6.68888888… -.68888888 ∙ 360 = About -248 Coterminal angle needs to be positive: 360 – 248 = 112o , Quadrant II The acute angle formed by the terminal side of the given angle and the x-axis. 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 Example 5: Find the measure of the reference angle for each angle. a. 240° Since 240o is between 180 and 270 the terminal side of the angle lies in Quadrant III. 240 – 180= 60o b. -305° 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. The reference angle is 55o.