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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 20º 20 360 380 Find a negative coterminal angle to 20º 20 360 340 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" 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 60º20'40" minutes 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 Converting from DD to DMS Express 50.525 in degrees, minutes, seconds To reverse the process, we multiply by 60 instead. 50º + .525(60) 50º + 31.5 50º + 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, let’s look at a circle. Definition: a radian is an arc length of one radius So, how many radians are there in a given circle? What’s the connection between degrees and radians? 6 4 θ 3 a little extra r 2 1 radian 360 2 r 360 180 r 57.3 2 180 We can use the two ratios to convert between radians and or 180 degrees. Example: Change 330˚ to radians: 330 11 180 6 In most cases, radians are left in terms of π 2 radians to degree measure. 3 2 180 120 3 Example: Convert 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. 5 128 180 11.2 cm Area of a sector (slice of pie): A = ½ r2θ (θ 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=½ r2θ 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 (ω): the speed at which an angle opens. Remember: θ is in radians. Ex. 6 rev/min, 360°/day, 2π 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 2π radians … so we’re talking about… 7.3 2 14.6 radians Let’s use the formula: t 14.6 9.2 rad / 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 vr 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