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) Initial Side Starting position Terminal SideEnding 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 20º Find a negative coterminal angle to 20º Types of questions you will be asked: Identify a) ALL angles coterminal with 45º, then b) find one positive coterminal angle and one negative coterminal angle. a) 45º + 360k (where k is any given integer). b) Some possible answers are 405º, 765º, - 315º, - 675º
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 60°20'40"
1 degree = 60 minutes 1° = 60 1 minute = 60 seconds 1 = 60 So … 1 degree = _________seconds 3600 Express 36 5010 as decimal degrees (DD) To complete the calculation, remember that … Converting from DMS to DD THEREFORE …
Try this: Converting DMS to DD 20 minutes.= 0.33333 (20/60) 40 seconds = 0.01111 (40/3600) Add up the degrees to get an answer: 60º + 0.33333 + 0.01111=60.34444 DD 60º20'40" degrees minutes seconds
Express 50.525 in degrees, minutes, seconds 50º +.525(60) 50º + 31.5 50º + 31 +.5(60) 50 degrees, 31 minutes, 30 seconds Converting from DD to DMS To reverse the process, we multiply by 60 instead.
So, what exactly is a RADIAN? Many math problems are more easily handled when degrees are converted to RADIANS. For a visual depiction of a radian, let’s look at a circle. θ 1 radian 2 3 4 5 6 a little extra r So, how many radians are there in a given circle? What’s the connection between degrees and radians? Definition: a radian is an arc length of one radius
We can use the two ratios to convert between radians and degrees. Example: Change 330˚ to radians: Example: Convert radians to degree measure. In most cases, radians are left in terms of π
Two formulas to know: 1.Arc Length of a circle: S = rθ (θ 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 = rθ 2.Area of a sector (slice of pie): A = ½ r 2 θ(θ in radians) Example: Find the area of a sector of the central angle measures radians and the radius of the circle is 16 inches. Round to nearest tenth. 11.2 cm A = ½ r 2 θ
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 rope around a pulley seat on a Ferris wheel 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 Angular Velocity (ω): the speed at which an angle opens. Definition: Remember: θ is in radians. Ex. 6 rev/min, 360 °/day, 2π rad/hour
Angular Velocity Example: determine the angular velocity if 7.3 revolutions are completed in 9 seconds. Round to nearest tenth. 1 revolution is 2π radians … so we’re talking about… Let’s use the formula:
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.
Linear Velocity Linear Velocity: the speed with which An object revolves a fixed distance from a central point. Definition: Ex. 55 mph, 6 ft/sec, 27 cm/min, 4.5 m/sec If you already know the angular velocity, then …
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. 5.5 ft/sec