More Trigonometry!! Section 4-2 Review Angles Standard Position Coterminal Angles Reference Angles Converting from Degrees degrees, minutes, seconds (DMS) Angle- formed by rotating a ray about its endpoint (vertex)

Terminal Side Ending position Initial Side Starting position Standard Position Initial side on positive x-axis and the vertex is on the origin An angle describes the amount and direction of rotation 120 210 Positive Angle- rotates counter-clockwise (CCW) Negative Angle- rotates clockwise (CW)

Coterminal Angles: Two angles with the same initial and terminal sides Find a positive coterminal angle to 20 20 360 380 Find a negative coterminal angle to 20 20 360 340 Types of questions you will be asked: Identify a) ALL angles coterminal with 45, then b) find one positive coterminal angle and one negative coterminal angle. a) 45 + 360k (where k is any given integer). b) Some possible answers are 405, 765, - 315, - 675 Decimal Degrees (DD)

Decimal degrees are similar to degrees/ minutes/seconds (DMS) except that minutes and seconds are expressed as decimal values. Decimal degrees make digital storage of coordinates easier and computations faster. 60.34444 instead of 6020'40" Converting from DMS to DD Express 365010as decimal degrees (DD) To complete the calculation, remember that 1 degree = 60 minutes 1 minute = 60 seconds

1 = 60 1 = 60 3600 So 1 degree = _________seconds THEREFORE Try this: Converting DMS to DD degrees seconds 6020'40" minutes

20 minutes.= 0.33333 (20/60) 40 seconds = 0.01111 (40/3600) Add up the degrees to get an answer: 60 + 0.33333 + 0.01111=60.34444 DD Converting from DD to DMS Express 50.525 in degrees, minutes, seconds To reverse the process, we multiply by 60 instead. 50 + .525(60) 50 + 31.5 50 + 31 + .5(60) 50 degrees, 31 minutes, 30 seconds

Homework Page 238 # 2 - 16 evens So, what exactly is a RADIAN? Many math problems are more easily handled when degrees are converted to RADIANS. 5 For a visual depiction of a radian, lets look at a circle. Definition: a radian is an arc length of

one radius So, how many radians are there in a given circle? Whats the connection between degrees and radians? 6 4 3 a little extra

r 2 1 radia n 360 2 r 360 180 r 57.3 2

180 or We can use the two ratios to convert between radians and 180 degrees. Example: Change 330 to radians: 330

11 180 6 In most cases, radians are left in terms of 2 radians to degree measure. 3 2 180 120 3

Example: Convert Two formulas to know: 1. Arc Length of a circle: S = r ( in radians) ( ( in radians) in radians)) Example: Given a central angle of 128 degrees, find the length of the intercepted arc in a circle of radius 5 centimeters. Round to nearest tenth. S = r ( in radians) 2.

5 128 11.2 cm 180 Area of a sector (slice of pie): A = r2 ( in radians) ( ( in radians) in radians)) Example: Find the area of a sector of the central angle measures the radius of the circle is 16 inches. Round to nearest tenth. A=r 2 1

5 2 16 335.1in 2 2 6 radians and Linear & Angular Velocity Things that turn have both a linear velocity and an angular velocity. Things that Turn - Examples tire on a car or bike buckets on a waterwheel

teeth on a gear can on a kitchen cabinet lazy susan propeller on an airplane horse on a Merry-Go-Round fins on a fan or a windmill earth on its axis Linear & Angular Velocity - Examples film on a projector or tape on a videotape turntable in a microwave oven blade on a lawnmower Earth around the sun seat on a Ferris wheel rope around a pulley a record on an old record player

drum/barrel in a clothes dryer Things that Turn - Examples lock on your locker hands on a clock roller brush on a vacuum cleaner tops & gyroscopes & dradle motor crankshaft blades in a blender roller skate wheels Carnival rides: tilt-a-whirl, scrambler, etc. weather vane washing machine agitator Angular Velocity Definition:

Angular Velocity ():): the speed at which an angle opens. Remember: is in radians. Ex. 6 rev/min, 360/day, 2 rad/hour t Angular Velocity Example: determine the angular velocity if 7.3 revolutions are completed in 9 seconds. Round to nearest tenth. 1 revolution is 2 radians so were talking about

7.3 2 14.6 radians Lets use the formula: t 14.6 9.2rad / sec 9sec Angular Velocity EXAMPLE 2: A carousel makes 2 5/8 rotations per minute. Determine the

angular velocity of a rider on the carousel in radians per second . 2 85 2.625revolutions 2.625revs 1min 2 radians radians 0.275 1min 60sec revolution sec Linear Velocity Definition:

Linear Velocity: the speed with which An object revolves a fixed distance from a central point. If you already know the angular velocity, then Ex. 55 mph, 6 ft/sec, 27 cm/min, 4.5 m/sec v r t r Linear Velocity

In the carousel scenario, one of the animals is 20 feet from the center. What is its linear velocity? Solution The cable moves at a fixed speed a linear velocity. r t .275radians 20 sec 5.5 ft/sec