13 2 General Angles and Radian Measure Definitions

13. 2 General Angles and Radian Measure

Definitions: � Initial side- a fixed ray used to generate an angle � Terminal side-a ray rotating about a vertex used to generate an angle. � Standard position- In a coordinate plan, the position of an angle whose vertex is at the origin and whose initial side is the positive xaxis

Picture 90˚ e sid al in m r Te 180˚ Initial side 270˚ 0˚

Examples: � Draw an angle with the given measure in standard position. Then tell in which quadrant the terminal side lies. ◦ -120˚ ◦ 400˚ ◦ 360˚

Definitions � Coterminal- two angles in standard position whose terminal sides coincide. ◦ Conterminal angles can be found by adding or subtracting multiples of 360˚

Examples: � Find one positive and one negative angle that are coterminal with ◦ -100˚ ◦ 575˚ ◦ -372˚

Definitions � Radian- the measure of an angle in standard position whose terminal side intercepts an arc of length r � Conversions Between Degrees and Radians ◦ To rewrite a degree measure in radians, multiply by ◦ To rewrite a radian measure in degrees, multiply by

Examples � Convert 320˚ to radians. � Convert -220˚ to radians. � Convert radians to degrees.

Examples � In a touring bicycle’s first gear, the chain passes over 32 teeth on the freewheel and 24 teeth on the chainwheel. If the chainwheel completes 3 rotations, through what angle does the freewheel turn? Give your answer in both degrees and radians.

Example: � In second gear, a bicycle’s chain passes over 26 teeth on the freewheel and 24 teeth on the chinwheel. If the chainwheel completes 13 rotations, through what angle does the freewheel turn? Give you answer in both degrees and radians.

Definitions: � Sector- a region that is bounded by two radii and an arc of the circle � Central angle- is the angle formed by the two radii � Arc Length and Area of a Sector �Arc Length: s=rθ �Area:

Examples: � Find the arc length and area of a sector with a radius of 5 centimeters and a central angle of 45˚ � Find the arc length and area of a sector with a radius of 12 inches and a central angle of 120˚.

Example: �A carousel with a diameter of 25 ft takes 16 sec to make one rotation. If you ride the carousel for 2 min, through what angle do you rotate? How many feet does a point on the outer edge of the carousel revolve?

Example: �A ferris wheel with a diameter of 50 ft takes 30 sec to make a rotation. If you ride the ferris wheel for 2 min 45 sec, through what angle do you rotate? If you seat is at the outer edge, how many feet do you rotate?