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More Trigonometry!! Section 4-2 Review Angles Standard Position Coterminal Angles Reference Angles Converting from Degrees – degrees, minutes, seconds (DMS)

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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

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An angle describes the amount and direction of rotation 120°–210° Positive Angle- rotates counter-clockwise (CCW) Negative Angle- rotates clockwise (CW)

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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

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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"

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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 …

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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

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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.

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Homework Page 238 # 2 - 16 evens

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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

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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

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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

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Linear & Angular Velocity Things that turn have both a linear velocity and an angular velocity.

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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

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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

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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

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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

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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:

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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.

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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 …

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Linear Velocity In the carousel scenario, one of the animals is 20 feet from the center. What is its linear velocity?

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Solution The cable moves at a fixed speed … a linear velocity. 5.5 ft/sec

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Angles and Their Measure Section 3.1. Objectives Convert between degrees, minutes, and seconds (DMS) and decimal forms for angles. Find the arc length.

Angles and Their Measure Section 3.1. Objectives Convert between degrees, minutes, and seconds (DMS) and decimal forms for angles. Find the arc length.

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