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4.1 Angles and Their Measures

4.1 Angles and Their Measures. By the end of today…. You should be able to: Convert between degrees and radians Find arc lengths Convert to nautical miles Solve problems using angular speed. Measuring Angles. Degrees Minutes Seconds (DMS).

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4.1 Angles and Their Measures

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  1. 4.1 Angles and Their Measures

  2. By the end of today….. You should be able to: • Convert between degrees and radians • Find arc lengths • Convert to nautical miles • Solve problems using angular speed

  3. Measuring Angles

  4. Degrees Minutes Seconds (DMS) • Degree: A unit of measurement for angles (1/180th of a straight angle) • Each degree is subdivided into 60 minutes (denote by ’) • Each minute is subdivided into 60 seconds (denoted by ”)

  5. Working with DMS Measure Convert from DMS to decimal form 118° 44’15” You Try! Convert from DMS to decimal form 23° 12’

  6. Working with DMS Measure Convert from decimal form to degrees, minutes, seconds 21.2° You Try! Convert from decimal form to degrees, minutes, seconds 118.32°

  7. What is a Radian? A central angle of a circle has measure 1 radian if it intercepts an arc with the same length as the radius.

  8. Let’s Explore! Constructing a 1-Radian Angle • Draw a large circle on a piece of paper using a compass • Identify and label the center (O) • Draw a horizontal radius from the Center (O) toward the right until your line intersects the circle. Call this point of intersection Point A. • Cut a piece of string the same exact size as the radius • Place one end of the string at A and bend it around the circle counterclockwise. Wherever that string ends, mark that as point B on your circle. • Draw the radius from the center to point B • The measure of angle AOB is one radian • How many times does it take for that piece of string to go entirely around the circle?

  9. Radians

  10. Degree-Radian Conversion • To convert from radians to degrees, multiply by: • To convert from degrees to radians, multiply by:

  11. You Try! Convert from degrees to radians • 60° • 52° You Try! Convert from radians to degrees • 4π/3 • π/10

  12. Arc Length (Radian Measure) (Degree Measure)

  13. Use the appropriate arc length formula to find the missing information

  14. The running lanes at Knightdale High School track are 1 meter wide. The inside radius of lane 1 is 33 meters and the inside radius of lane 2 is 34 meters. Henry, TyQuan, and Quez are obsessed with Precalculus and want to know how much longerlane 2 is than lane 1 around one turn?

  15. Refer to the 16 compass bearings shown. North corresponds to an angle of 0°, and other angles are measured clockwise from north. • Find the angle in degrees that describes the compass bearing: • NE (northeast) • NNE (north-northeast) • WSW (west-southwest)

  16. Angular and Linear Motion A radial arm saw has a circular cutting blade with a diameter of 10 inches. It spins at 2000 rpm. If there are 12 cutting teeth per inch on the cutting blade, how many teeth cross the cutting surface each second?

  17. Nautical and Statute Miles • A nautical mile: the length of 1 minute of arc along the Earth’s equator • Statute miles: the familiar “land mile” of 5280 feet Distance Conversions: • 1 nautical mile = ≈ 1.15 statute mile • 1 statute mile= ≈ .87 nautical mile Ex) Points C and D are 895 statute miles apart. How far apart are C and D in nautical miles?

  18. Homework Pg. 358-359 4, 8, 12, 16, 22, 26, 28, 30,34, 43, 49, 50

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