Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Radian and Degree Measure MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 J. Robert Buchanan Radian and Degree Measure Objectives In this lesson we will learn to: describe angles, use radian measure, use degree measure, use angles to model and solve real-world problems. J. Robert Buchanan Radian and Degree Measure Background We now begin a study of trigonometry (Greek for “measurement of triangles”). Terminal Side Vertex Initial Side J. Robert Buchanan Radian and Degree Measure Standard Position If the vertex is the origin and the initial side lies along the positive x-axis, the angle is said to be in standard position. y Terminal Side x Initial Side J. Robert Buchanan Radian and Degree Measure Orientation and Notation Angles will be named by uppercase letters (A, B, C, . . . ) or Greek letters (α, β, γ, . . . ). An angle is positive if it is generated by rotating the terminal side counterclockwise (abbreviated ccw) from the initial side. An angle is negative if it is generated by rotating the terminal side clockwise (abbreviated cw) from the initial side. Angles that have the same initial and terminal sides are called coterminal angles. J. Robert Buchanan Radian and Degree Measure Radian Measure Definition One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle. If θ is the radian measure of the angle then θ= s . r Recall: the circumference of a circle is C = 2πr , thus the radian measure of one complete revolution is θ= 2πr = 2π ≈ 6.28. r J. Robert Buchanan Radian and Degree Measure Illustration y r r=s Θ x r J. Robert Buchanan Radian and Degree Measure Common Angles (in radians) Π6 Π4 Π2 Π Π3 3Π2 J. Robert Buchanan Radian and Degree Measure Angles and Quadrants Θ=Π2 Quadrant II Quadrant I Π2<Θ<Π 0<Θ<Π2 Θ=Π Θ=0 Π<Θ<3Π2 3Π2<Θ<2Π Quadrant III Quadrant IV Θ=3Π2 J. Robert Buchanan Radian and Degree Measure Coterminal Angles Recall: two angles are coterminal if they have the same initial and terminal sides. Since there are 2π radians in a complete revolution, coterminal angles will have radian measures which differ by an integer multiple of 2π. Angles α and β are coterminal if they are in standard position and α = β + 2n π for some integer n. J. Robert Buchanan Radian and Degree Measure Examples Find three coterminal angles to each of the following. π θ=− 6 2π θ= 3 5π θ=− 4 J. Robert Buchanan Radian and Degree Measure Complementary and Supplementary Angles Definition Two positive angles α and β are complementary if their sum is π/2. Two positive angles α and β are supplementary if their sum is π. J. Robert Buchanan Radian and Degree Measure Complementary and Supplementary Angles Definition Two positive angles α and β are complementary if their sum is π/2. Two positive angles α and β are supplementary if their sum is π. Example If possible, find the complement and the supplement to each of the following angles. 3π θ= 7 θ= 2π 3 J. Robert Buchanan Radian and Degree Measure Complementary and Supplementary Angles Definition Two positive angles α and β are complementary if their sum is π/2. Two positive angles α and β are supplementary if their sum is π. Example If possible, find the complement and the supplement to each of the following angles. 3π θ= 7 π 4π Complement: , Supplement: 14 7 2π θ= 3 J. Robert Buchanan Radian and Degree Measure Complementary and Supplementary Angles Definition Two positive angles α and β are complementary if their sum is π/2. Two positive angles α and β are supplementary if their sum is π. Example If possible, find the complement and the supplement to each of the following angles. 3π θ= 7 π 4π Complement: , Supplement: 14 7 2π θ= 3 π Complement: none, Supplement: 3 J. Robert Buchanan Radian and Degree Measure Degree Measure Degrees are another way to measure angles. A degree is equivalent to a rotation of 1/360 of a complete revolution. Conversions 1 2 π rad . 180◦ 180◦ To convert radians to degrees, multiply radians by . π rad To convert degrees to radians, multiply degrees by J. Robert Buchanan Radian and Degree Measure Illustration 90° ° 60° 120 135° 45° 150° 30° 180° 0° 210° 330° 225° 315° 240° 300° 270° J. Robert Buchanan Radian and Degree Measure Examples Complete the angle measures in the following table. Angle α β γ θ Radian Measure Degree Measure 15◦ π/3 135◦ 5π/3 J. Robert Buchanan Radian and Degree Measure Examples Complete the angle measures in the following table. Angle α β γ θ Radian Measure π/12 π/3 3π/4 5π/3 J. Robert Buchanan Degree Measure 15◦ 60◦ 135◦ 300◦ Radian and Degree Measure Application: Arc Length Arc Length For a circle of radius r , a central angle θ intercepts an arc of length s given by s = rθ where θ is measured in radians. Note that if r = 1, then s = θ, and the radian measure of θ equals the arc length. J. Robert Buchanan Radian and Degree Measure Example A circle has a radius of 5 feet. Find the length of the arc intercepted by a central angle of 5π/4 120◦ J. Robert Buchanan Radian and Degree Measure Example A circle has a radius of 5 feet. Find the length of the arc intercepted by a central angle of 5π/4 s = (5) 5π 25π = ≈ 19.635 4 4 feet 120◦ J. Robert Buchanan Radian and Degree Measure Example A circle has a radius of 5 feet. Find the length of the arc intercepted by a central angle of 5π/4 s = (5) 5π 25π = ≈ 19.635 4 4 feet 120◦ s = (5)(120◦ ) π 10π = ≈ 10.472 180◦ 3 J. Robert Buchanan Radian and Degree Measure feet Application: Linear and Angular Speeds Linear and Angular Speeds Consider a particle moving at a 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 v of the particle is Linear speed v = s arc length = . time t Moreover, if θ is the angle (in radian measure) corresponding to the arc length s, then the angular speed ω (lowercase Greek letter omega) of the particle is Angular speed ω = J. Robert Buchanan central angle θ = . time t Radian and Degree Measure Example A carousel with a 50-foot diameter makes 4 revolutions per minute. 1 Find the angular speed of the carousel in radians per minute. 2 Find the linear speed (in feet per minute) of the platform rim of the carousel. J. Robert Buchanan Radian and Degree Measure Example A carousel with a 50-foot diameter makes 4 revolutions per minute. 1 Find the angular speed of the carousel in radians per minute. ω = 4 rev/min = (4 rev/min)(2π rad/rev) = 8π rad/min 2 Find the linear speed (in feet per minute) of the platform rim of the carousel. J. Robert Buchanan Radian and Degree Measure Example A carousel with a 50-foot diameter makes 4 revolutions per minute. 1 Find the angular speed of the carousel in radians per minute. ω = 4 rev/min = (4 rev/min)(2π rad/rev) = 8π rad/min 2 Find the linear speed (in feet per minute) of the platform rim of the carousel. v = r ω = (25)(8π) = 200π ≈ 828.32 feet/min J. Robert Buchanan Radian and Degree Measure Application: Area of a Sector A sector of a circle is the region bounded between the two radii of an intercepted arc. Θ r A= J. Robert Buchanan 1 2 r θ 2 Radian and Degree Measure Example A car’s rear windshield wiper rotates 125◦ . The total length of the wiper mechanism is 25 inches and the blade wipes the windshield over a distance of 14 inches. Find the area covered by the wiper. J. Robert Buchanan Radian and Degree Measure Example A car’s rear windshield wiper rotates 125◦ . The total length of the wiper mechanism is 25 inches and the blade wipes the windshield over a distance of 14 inches. Find the area covered by the wiper. First convert the angle to radian measure. 125◦ = 125π 25π = 180 36 J. Robert Buchanan Radian and Degree Measure Example A car’s rear windshield wiper rotates 125◦ . The total length of the wiper mechanism is 25 inches and the blade wipes the windshield over a distance of 14 inches. Find the area covered by the wiper. First convert the angle to radian measure. 125π 25π = 180 36 1 25π 1 25π A = (25)2 − (11)2 2 36 2 36 = 175π ≈ 549.779 in2 125◦ = J. Robert Buchanan Radian and Degree Measure Homework Read Section 4.1. Exercises: 1, 5, 9, 13, . . . , 113, 117 J. Robert Buchanan Radian and Degree Measure