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Angles and Their Measure 1.1 Objectives Students will be able to describe angles. Students will be able to use radian measure. Students will be able to us degree measure. Students will be able to use angles to model and solve real-life problems. Angles An angle is formed by two rays that have a common endpoint called the vertex. One ray is called the initial side and the other the terminal side. The arrow near the vertex shows the direction and the amount of rotation from the initial side to the terminal side. C A è Terminal Side Initial Side B Vertex Angles of the Rectangular Coordinate System An angle is in standard position if • its vertex is at the origin of a rectangular coordinate system and • its initial side lies along the positive x-axis. y y is positive Terminal Side Vertex Initial Side x x Vertex Terminal Initial Side Side is negative Positive angles rotate counterclockwise. Negative angles rotate clockwise. Coterminal Angles An angle of xº is coterminal with angles of xº + k · 360º where k is an integer. The initial and terminal sides are the same. Definition of a Radian • One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle. Radian Measure Consider an arc of length s on a circle or radius r. The measure of the central angle that intercepts the arc is s = s/r radians. The circumference is r 2πr units O r See Pg. 135 Finding Complements and Supplements • Angles are complementary if their sum is π/2. • Angles are supplementary if their sum is π. • Pg. 142 #18a, 20a, 22a, 24, 30, 32a, 34a • Measuring Angles Using Degrees The figures below show angles classified by their degree measurement. An acute angle measures less than 90º. A right angle, one quarter of a complete rotation, measures 90º and can be identified by a small square at the vertex. An obtuse angle measures more than 90º but less than 180º. A straight angle has measure 180º. 90º 180º Acute angle Right angle Obtuse angle Straight angle 0º < < 90º 1/4 rotation 90º < < 180º 1/2 rotation Conversion between Degrees and Radians • Using the basic relationship • radians = 180º • 2π radians = 360º = (one complete rotation) • To convert degrees to radians, multiply degrees by ( radians) / 180 • To convert radians to degrees, multiply radians by 180 / ( radians) Example Convert each angle in degrees to radians 40º 75º -160º Example cont. Solution: 2 40 40 180 9 5 75 75 180 12 8 8 180 o 160o 9 9 rads Length of a Circular Arc Let r be the radius of a circle and the non- negative radian measure of a central angle of the circle. The length of the arc intercepted by the central angle is s=r s Pg. 144 # 90, 96, 98 O r Area of a Sector of a Circle For a circle of radius r, the area A of a sector of the circle with central angle (in radians) is given by 1 2 A r s 2 r Pg. 144 #102 Angular Speed Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc travelled in time t, then arch length s Linear Speed time t If Ѳ is the angle measure corresponding to arc length s then Angular Speed central angle time t • Groups: Pg 143 – 145 # 18b, 22b, 32b, 44, 56, 70, 76, 92, 104, 112 • Homework: 17 – 33 odd, 41 – 79 odd, 89 – 103 odd, 111