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Precalculus Lesson 4.1 Radian and Degree Measure Important Vocabulary Trigonometry: The Greek word for “measurement of triangles.” Central angle of a circle: An angle whose vertex is the center of the circle. Complementary angles: Two positive angles whose sum is p/2 radians or 90°. Supplementary angles: Two positive angles whose sum is p radians or 180°. Degree: The most common unit of angle measure, denoted by the symbol °. A measure of one degree (1°) is equivalent to a rotation of 1/360 of a complete revolution about the vertex of an angle. Prerequisite Knowledge: •An angle is two rays with the same initial point. •The measure of an angle is the amount of rotation required to rotate one side, called the initial side, to the other side, called the terminal side. •The shared initial point of the two rays is called the vertex of the angle. •An angle is in standard position if its vertex is at the origin of the rectangular coordinate system and the initial side lies along the positive x-axis. Positive and Negative Angles •If the rotation of the angle is in the counterclockwise direction, then the angle is said to be positive. If the rotation is clockwise, then the angle is negative. Radian Measure The measure of an angle is determined by the amount of rotation from the initial side to the terminal side. One radian is the measure of a central angle ѳ that intercepts an arc s equal in length to the radius r of the circle. There are 2π radians of rotation in an entire circle. Types of Angles Angles with measure between 0 and π/2 are called acute. Angles with measure between π/2 and π are called obtuse. Two positive angles are said to be complementary if the sum of their measures is π/2, and supplementary if the sum of their measures is π. Example 1. Find the complement and supplement of a π/5 angle. Coterminal Angles •Two angles in standard position that have the same terminal side are said to be coterminal. Coterminal Angles To find an angle that is coterminal to a given angle ѳ,add or subtract 2π or integer multiples of 2π to the measure of ѳ. Example 2: Find an angle that is coterminal with ѳ = - π/8. Degree Measure Since there are 360°, or 2π radians, in an entire circle, 360° = 2π radians 80° = π radians 1° = (π/180) rad 1 rad = 180°/π To convert degrees to radians, multiply by (π/180) rad To convert radians to degrees, multiply by 180°/π Converting Degrees and Radians Example 3: Convert the following degree measures to radian measure. a) 120 b) −315 Example 4: Convert the following radian measures to degrees. a) 5π/6 b) 7 More Conversion Degrees °, Minutes ‘, and Seconds “ 1’ = one minute = 1/60° 1” = one second = 1/3600° 64 degrees, 32 minutes, and 47 seconds = 64°32’47” Decimal to DMS: Enter the decimal in the calculator. Press `A4e DMS to Degrees: Enter the number in DMS form. Then press e Linear and Angular Speed With radian measure: θ = s/r where s is the arc length so s = rθ Consider a particle moving at constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed of the particle is: linear speed = (arc length)/(time) = s/t = rθ/t If ѳ is the angle (in radian measure) corresponding to the arc length s, then the angular speed of the particle is angular speed = (central angle)/(time) = ѳ/t = s/rt Example 6: Find the length of the arc that subtends a central angle with measure 120° in a circle with radius 5 inches. Example 7: A 6-inch-diameter gear makes 2.5 revolutions per second. Find the angular speed of the gear in radians per second. 5p radians per second Example 8: An automobile is traveling at 65 mph. If each tire has a radius of 15 inches, at what rate are the tires spinning in revolutions per minute (rpm)? Homework for Lesson 4.1 Yellow Book (Period 4): p. 265-268 # 3-19 odd, 31, 33, 39-73 odd, 77-87 odd Orange Book (Period 7): p. 255-258 #5-21 odd, 35, 37, 45-87 odd