The Beginnings of Chapter 4 Section 4 1

The Beginnings of Chapter 4 Section 4. 1 a: Angles, their measures, and arc length

First, Some Definitions Degree – represented by the symbol , is a unit of angular measure equal to 1/180 th of a straight angle. Note: Each degree is subdivided into 60 minutes (denoted by ’ ), and each minute is subdivided into 60 seconds (denoted by ” ). This is called the DMS form for angle measure.

Conversion Problems Convert each of the following from DMS to decimal form. Each minute is 1/60 th of a degree, and each second is 1/3600 th of a degree:

Conversion Problems Convert the following from decimal to DMS form. We need to convert the fractional part to minutes and seconds. First, convert the 0. 425 degrees to minutes: Then, convert 0. 5 minutes to seconds: Final Answer:

Radians So, what’s the problem with degrees, anyway? ? ? Degree units have no mathematical relationship whatsoever to linear units!!! EX: There are 360 degrees in a circle with radius 1… What relationship does the 360 have to the 1? ? ? Is it 360 times as big? ? ? Enter RADIANS to help solve these dilemmas…

Definition: Radians A central angle of a circle has measure 1 radian if it intercepts an arc with the same length as the radius. a 1 radian a

Degree-Radian Conversion To convert radians to degrees, use the conversion factor: Note: radians and 180 both measure a straight angle!!! We will use dimensional analysis to convert all angles.

Guided Practice How many radians are in 90 degrees? radians How many degrees are in radians? radians

More Definitions 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. Ex: Sketch a diagram of the path of a boat leaving a harbor with a bearing of 155 Harbor 155 Path of boat

ARC LENGTH FORMULA (RADIAN MEASURE) If 0 is a central angle in a circle of radius r, and if 0 is measured in radians, then the length s of the intercepted arc is given by

ARC LENGTH FORMULA (DEGREE MEASURE) If 0 is a central angle in a circle of radius r, and if 0 is measured in degrees, then the length s of the intercepted arc is given by

GUIDED PRACTICE Use the appropriate arc length formula to find the missing information. cm cm ft in m rad rad

GUIDED PRACTICE Find the perimeter of a 60 slice of a large (7 in. radius) pizza. Delicious slice of pizza: Perimeter: 7 in + s in 7 in Find s : 60 7 in The perimeter is approximately 21. 330 inches

GUIDED PRACTICE The running lanes at a certain track are 1 meter wide. The inside radius of lane 1 is 33 meters and the inside radius of lane 2 is 34 meters. How much longer is lane 2 than lane 1 around one turn? Lane 2 Lane 1 33 m Each lane is a semicircle with central angle and length. Therefore, the difference in their lengths is 34 m Lane 2 is about 3. 142 meters longer than lane 1.

GUIDED PRACTICE It takes ten identical pieces to form a circular track for a pair of toy racing cars. If the inside arc of each piece is 3. 4 inches shorter than the outside arc, what is the width of the track? What is the measure of each of these central angles? Inside arc length for one piece of track: Outside arc length for one piece of track:

GUIDED PRACTICE It takes ten identical pieces to form a circular track for a pair of toy racing cars. If the inside arc of each piece is 3. 4 inches shorter than the outside arc, what is the width of the track? But we were given the difference between these arc lengths: inches

GUIDED PRACTICE The concentric circles on an archery target are 6 inches apart. The inner circle (red) has perimeter of 37. 7 inches. What is the perimeter of the next-largest (yellow) circle? Perimeter of inner circle: Perimeter of outer circle: 6 d 6 inches

Whiteboard Conversion Problems Convert each of the following from DMS to decimal form.

Whiteboard Conversion Problems Convert each of the following from decimal to DMS form. Convert the 0. 98 degrees to minutes: Convert the 0. 29 degrees to minutes: Convert the 0. 8 minutes to seconds: Convert the 0. 4 minutes to seconds:

Whiteboard Practice Convert each of the following from DMS to radians. (a) (b)

Whiteboard Practice Convert each of the following from radians to degrees. (a) (b)

WHITEBOARD PRACTICE Convert from DMS to radians rad Convert from radians to degrees