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Exploring Angles and Arcs in Geometry

Discover the fundamentals of angles, degree measures, and coterminal angles in geometry. Learn to convert between degrees and radians, calculate complementary and supplementary angles, and find measures of coterminal angles.

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Exploring Angles and Arcs in Geometry

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  1. 5.1 Angles and Arcs • Basic Terminology • Two distinct points A and B determinethe line AB. • The portion of the line including the points A and B is the line segment AB. • The portion of the line that starts at A and continues through B is called ray AB. • An angle is formed by rotating a ray, the initial side, around its endpoint, the vertex, to a terminal side.

  2. 5.1 Degree Measure • Degree Measure • Developed by the Babylonians around 4000 yrs ago. • Divided the circumference of the circle into 360 parts. One possible reason for this is because there are approximately that number of days in a year. • There are 360° in one rotation. • An acute angle is an angle between 0° and 90°. • A right angle is an angle that is exactly 90°. • An obtuse angle is an angle that is greater than 90° but less than 180°. • A straight angleis an angle that is exactly 180°.

  3. 5.1 Finding Measures of Complementary and Supplementary Angles • If the sum of two positive angles is 90°, the angles are called complementary. • If the sum of two positive angles is 180°, the angles are called supplementary. Example Find the measure of each angle in the given figure. (a) (b) (Supplementary angles) (Complementary angles) Angles are 60 and 30 degrees Angles are 72 and 108 degrees

  4. 5.1 Calculating With Degrees, Minutes, and Seconds • One minute, written 1', is of a degree. • One second, written 1", is of a minute. Example Perform the calculation Solution ' Since 75' = 1° + 15', the sum is written as 84°15'.

  5. 5.1 Converting Between Decimal Degrees and Degrees, Minutes, and Seconds Example (a)Convert 74º814 to decimal degrees. (b) Convert 34.817º to degrees, minutes, and seconds. Analytic Solution (a)Since

  6. 5.1 Converting Between Decimal Degrees and Degrees, Minutes, and Seconds (b) Graphing Calculator Solution

  7. 5.1 Coterminal Angles • Quadrantal Angles are angles in standard position (vertex at the origin and initial side along the positive x- axis) with terminal sides along the x or y axis, i.e. 90°, 180°, 270°, etc. • Coterminal Angles are angles that have the same initial side and the same terminal side.

  8. 5.1 Finding Measures of Coterminal Angles Example Find the angles of smallest possible positive measure coterminal with each angle. (a) 908° (b) –75° Solution Add or subtract 360°as many times as needed to get an angle between 0° and 360°. (a) 908 – 2(360) = 188 degrees (b) 360 + (-75) = 285 degrees • Let n be an integer, we have an infinite number of coterminal angles: e.g. 60° + n· 360°.

  9. 5.1 Radian Measure An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian. • The radian is a real number, where the degree is a unit of measurement. • The circumferenceof a circle, given by C = 2 r, where r is the radius of the circle, shows that an angle of 360º has measure 2 radians.

  10. 5.1 Converting Between Degrees and Radians • Multiply a radian measure by 180º/ and simplify to convert to degrees. For example, • Multiply a degree measure by  /180º and simplify to convert to radians. For example,

  11. 5.1 Converting Between Degrees and Radians With the Graphing Calculator Example Convert 249.8º to radians. Solution Put the calculator in radian mode. Example Convert 4.25 radians to degrees. Solution Put the calculator in degree mode.

  12. 5.1 Equivalent Angle Measures in Degrees and Radians Figure 18 pg 9

  13. 5.1 Arc Length The length s of the arc intercepted on a circle of radius r by a central angle of measure  radians is given by the product of the radius and the radian measure of the angle, or s = r,  in radians Example A circle has a radius of 25 inches. Find the length of an arc intercepted by a central angle of 45º. Solution

  14. 5.1 Linear and Angular Speed • Angular speed (omega) measures the speed of rotation (angle generated in one unit of time) and is defined by • Linear speed (the distance travelled per unit of time)  is defined by • Since the distance traveled along a circle is given by the arc length s, we can rewrite  as

  15. 5.1 Finding Linear Speed and Distance Traveled by a Satellite Example A satellite traveling in a circular orbit 1600 km above the surface of the Earth takes two hours to complete an orbit. The radius of the Earth is 6400 km. • Find the linear speed of the satellite. • Find the distance traveled in 4.5 hours. Figure 24 pg 12

  16. 5.1 Finding Linear Speed and Distance Traveled by a Satellite Solution (a) The distance from the Earth’s center is r = 1600 + 6400 = 8000 km. For one orbit,  = 2, so s = r = 8000(2) km. With t = 2 hours, we have (b) s = t = 8000(4.5)  110,000 km

  17. Area of a Circular Sector • A = • must be in radians

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