Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download radians or 90 degrees.
Document related concepts
Transcript
P.o.D. 1.) Divide 6π₯ 3 β 4π₯ 2 by 2π₯ 2 + 1. 2.) Solve the inequality 1 π₯+1 β₯ 1 . π₯+5 3.) Find ALL the zeros of π(π₯ ) = 2π₯ 4 β 11π₯ 3 + 30π₯ 2 β 62π₯ β 40 1.) 3π₯ β 2 β 3π₯β2 2π₯ 2 +1 2.) π₯ < β5 or x> -1 3.) 4, β1/2, 1+3i, 1-3i 4.1 β Radian and Degree Measure Learning Target: Be able to describe angles in both radians and degrees. *Chapter 4 begins the study of Trigonometry β measures of Triangles. Angles have two sides: 1. An initial side 2. A terminal side Terminal Side Initial Side - The endpoint of the two rays is known as the vertex. - An angle centered at the origin is said to be in Standard Position. - Positive angles are measured counterclockwise. - Negative angles are measure clockwise. - If two angles have the same position, then they are said to be coterminal. Radian vs. Degree: - Just as distance may be measured in feet and centimeters, angles can be measured in both radians and degrees. Definition of a Radian: π π = , where s is the arc length and r is π the radius. Conversion Factors: 2π πππππππ = 360° π πππππππ = 180° 1 ππππππ β 57.3° Some Other Common Radian Measures: π 45° = πππππππ 4 π 60° = πππππππ 3 π 30° = πππππππ 6 π 90° = πππππππ 2 Acute Angles are between 0 and π 2 radians. π Obtuse Angles are between and π 2 radians. EX: For the positive angle 9π 4 subtract 2π to obtain a coterminal angle. 9π 9π 8π π β 2π = β = 4 4 4 4 EX: For the positive angle 5π 6 subtract 2π to obtain a coterminal angle. 5π 5π 12π 7π β 2π = β =β 6 6 6 6 EX: For the negative angle β 3π 4 , add 2π to find a coterminal angle. 3π β3π 8π 5π β + 2π = + = 4 4 4 4 Recall your Quadrants for Geometry: In Q1 ο 0 < π < In Q2 ο π 2 π 2 <π<π In Q3 ο π < π < In Q4 ο 3π 2 3π 2 < π < 2π Complementary β two angles whose π sum is radians or 90 degrees. 2 Supplementary β two angles whose sum is π radians or 180 degrees. EX: Find the complement and π supplement of . 6 Complement: π π 3π π β = β 2 6 6 6 2π π = = 6 3 Supplement: π 6π π πβ = β 6 6 6 5π = 6 EX: Find the complement and supplement of 5π 6 . Complement: π 5π β = 2 6 3π 5π β = 6 6 β2π βπ = 6 3 Since the angle is negative, no complement exists. Supplement: 5π πβ 6 6π 5π = β 6 6 π = 6 *There are 360 degrees in a circle. Conversions Between Degrees and Radians: 1. To convert degrees to radians, multiply degrees by π . 180 2. To convert radians to degrees, multiply radians by 180 π . EX: Convert 60 degrees to radians in terms of pi. 60° π πππππππ × = 1 180° 60π 180 π = πππππππ 3 EX: Convert 320 degrees to radians in terms of pi. 320 π × 1 180 320π = 180 16π = πππππππ 9 EX: Convert -30 degrees to radians in terms of pi. β30 π × 1 180 β30π = 180 βπ = πππππππ 6 *We could write this as a positive angle. βπ βπ 12π + 2π = + 6 6 6 11π = πππππππ 6 π EX: Express as a degree measure. 6 π 180 × 6 π 180 = 6 = 30° EX: Express 5π 3 as a degree measure. 5π 180 × 3 π 900 = 3 = 300° EX: Express 3 radians as a degree measure. 3 πππ 180° × 1 π πππ 540 = πππππππ π β 171.887° *We could also have done 3(57.3) = 171.9° *Letβs write a calculator program to convert radians to degrees and vice versa. Recall: π = Arc Length: π π π = ππ, where r is the radius and theta is the measure of the central angle. - It is important to note that theta must always be in radians when used in a formula. EX: A circle has a radius of 27 inches. Find the length of the arc intercepted by a central angle of 160 degrees. First, convert 160 degrees to radian measure. 160° π × 1 180° 160π = 180 8π = 9 Next, apply the formula for arc length, π = ππ. 8π π = 27 ( ) 9 = 24π β 75.398 πππβππ Linear Speed (v): Linear Speed v = πππ πππππ‘β π‘πππ = π π‘ Angular Speed π (omega): πβππππ ππ π‘βπ ππππ‘πππ πππππ π π= = π‘πππ π‘ *Study and memorize the tan box in the lower left hand corner of page 287. EX: The second hand of a clock is 8 centimeters long. Find the linear speed of the tip of this second hand as it passes around the clock face. We first need to find the distance (arc length) traveled by the second hand as it makes one complete revolution. π = ππ = 8(2π) = 16π Now find its linear speed. π 16π π£= = π‘ 60 π ππππππ 4π ππ βπ = 15 β 0.8378 ππ/π EX: The circular blade on a saw rotates at 2400 revolutions per minute. Find the angular speed in radians per second. We can use a process known as dimensional analysis. 2400 πππ£ 2π πππ 1 πππ × × 1 πππ 1 πππ£ 60 π ππ 4800π πππ = 60 π ππ = 80π πππ/π ππ EX: Referring to the previous problem, the blade has a radius of 4 inches. Find the linear speed of a blade tip in inches per second. Linear Velocity is Angular Velocity multiplied by Radius, π£ = ππ π£ = 80π(4) = 320π β 1005.3096 ππ/π ππ Area of a Sector: 1 2 π΄= π π 2 EX: A sprinkler on a golf course is set to spray water over a distance of 75 feet and rotates through an angle of 135 degrees. Find the area of the fairway watered by the sprinkler. Letβs first draw a picture of the situation. 135° π πππ π= × 1 180° 3π = πππ. 4 1 3π 2 π΄ = (75) ( ) 2 4 16875π = 8 β 6626.797 π ππ’πππ ππππ‘ *Letβs write a calculator program to find the area of a sector. Upon completion of this lesson, you should be able to: 1. Convert from radians to degrees and vice versa. 2. Find the supplement and/or complement of an angle. 3. Find the arc length of a sector. 4. Compute linear and angular speed. 5. Find the area of a sector. For more information, visit http://academics.utep.edu/Portals/1788/CALCULUS%20 MATERIAL/4_1%20RADIAN%20N%20DEGREES%20MEAS URES.pdf HW Pg.290 6-90 6ths, 102, 106, 117120.
