DAY 19 ANGLES Objective: SWBAT convert angles between radians, degrees and DMS. SWBAT solve problems involving Arc Length, Angular and Linear Motion EQ: How do we convert angles from degree to radian? Radian to degree? How is arc length measured? What is the difference between angular and linear motion? Classwork/Home Learning: p. 325 # 2, 4, 6, 8, 12, 16, 21, 22, 26, & 31 /p. 325 #1, 5, 7, 9-12, 17-20, 25 & 27

4-1 Angles and Their Measures

MEASURING ANGLES Angles can also be measured in RADIANS! ANGLES can be measured in decimal degree OR DMS DMS works just like time! There are 60 minutes in a Degree, and 60 Seconds in each minute. Angles can also be measured in RADIANS! PI radians is equivalent to 180 degrees or a straight angle.

EXAMPLE: Decimal Degree (dd) to DMS 67.42 means 67 and .42 of a whole degree which = 60 minutes OR .42*60=25.2 25.2 minutes means 25 minutes and .2*60 seconds OR 12 seconds 25.2 minutes

Example: DMS to Decimal Degree In order to convert back into Decimal Degree. Just Take Your Minutes/60 + Seconds/3600 12/3600+25/60=.42 OR 67.42 Degrees

Let’s try a few: #2 #4 #6 #8

Calculator Short-Cut! Your calculator probably has built-in functional- ity to convert degrees to DMS. You can also type in an angle in DMS (Use the symbols within the ANGLE menu and ENTER to convert it back to Decimal Degree

An Application: The course OR Bearing In navigation, the course or bearing of an object is sometimes given as the angle of the line of travel measured clockwise from due north. For example, the line of travel pictured here has the bearing of 155°.

Radian measure—a natural way! Radian measure, as opposed to the degree of an angle, should probably be credited to Roger Cotes in 1714. He had the radian in everything but name, and he recognized its naturalness as a unit of angular measure. The term “radian” first appeared in print in 1873, at Queen's College, Belfast. In calculus and most other branches of mathematics beyond practical geometry, angles are universally measured in radians.

Radian Measure An angle can be represented by an ‘arc length” Every Circle has a circumference of 2pi * radius. If we let the radius become the unit of measure, then every circle = 2 pi radians. Every angle can be re-written as a piece of pi!

It’s All About….. How many radians are in 90 degrees? Since radians and 180° both measure a straight angle, we can use the conversion factor ( /(180°) = 1 to convert degrees to radians: (b) How many degrees are in /3 radians?

CIRCULAR ARC LENGTH: Finding ARC lengths (s) If “theta” is a central angle in a circle of radius r, and “theta” is measured in radians, then the length s of the intercepted arc is given by

CIRCULAR ARC LENGTH Find the length of an arc intercepted by a central angle of 1/2 radian in a circle of radius 5 in. Find the perimeter of a 60° slice of a large (7-in. radius) pizza

ANGULAR VS. LINEAR MOTION Angular speed is (measured in units like revolutions per minute. Linear speed is measured in units like miles per hour.

EXAMPLE: LINEAR MOTION Albert Juarez’s truck has wheels 36 in. in diameter. If the wheels are rotating at 630 rpm (revolutions per minute), find the truck’s speed in miles per hour.

NAUATICAL MILES A nautical mile (naut mi) is the length of 1 minute of arc along Earth’s equator. The figure shows, though not to scale, a central angle AOB of Earth that measures 1/60 of a degree. It intercepts an arc 1 naut mi long. The arc length formula allows us to convert between nautical miles and statute miles (stat mi), the familiar “land mile” of 5280 ft.

NAUTICAL MILES Navigation Points A and B are 257 naut mi apart. How far apart are A and B in statute miles? Although Earth is not a perfect sphere, its diameter is, on average, 7912.18 stat mi. A nautical mile is 1′ of Earth’s circumference at the equator.

Let’s Apply It—Now You Try It!