Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Homework, Page 356 Convert from DMS to decimal form. 1. 2312 12 2312 23 28.2 60 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 1 Homework, Page 356 Convert from decimal form to DMS. 5. 21.2 21.2 2112 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 2 Homework, Page 356 Convert from decimal or DMS to radians. 9. 60 60 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 3 Homework, Page 356 Convert from decimal or DMS to radians. 13. 71.72 71.72 1.2518 rad Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 4 Homework, Page 356 Convert from radians to degrees. 17. 6 180 * 30 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 5 Homework, Page 356 Convert from radians to degrees. 7 21. 9 7 180 * 140 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 6 Homework, Page 356 Use the appropriate arc length formula to find the missing information. r 25. s ? 2 in 25 rad s s r 2* 25 50 in r Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 7 Homework, Page 356 Use the appropriate arc length formula to find the missing information. 29. s r 3m 1m ? s 3 3 rad r 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 8 Homework, Page 356 A central angle θ intercepts arcs s1 and s2 on two concentric circles with radii r1 and r2, respectively. Find the missing information. r1 s1 r2 s2 33. ? 11 cm 9 cm 44 cm ? s 9 0.818 r 11 1 1 s s 9 s s 36 cm r r 11 44 1 2 2 2 1 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 9 Homework, Page 356 37. It takes ten identical pieces to form a circular track for a pair of toy racing cars. If the inside arc of each piece is 3.4 inches shorter than the outside arc, what is the width of the track? s s 2 s 3.4 s r s 3.4 rs r r 10 r r 1 2 2 2 2 1 2 1 2 1 2 2 r s 3.4r rs 3.4r r s rs 3.4r r r s 2 2 2 1 2 2 2 2 1 2 2 2 1 2 3.4r 10 r r 3.4* 1.7 r r 5.341 in s 2 2 2 1 2 1 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 10 Homework, Page 356 41. Which compass bearing is closest to a bearing of 121º? E 090 SE 135 ESE 112.5 East-southeast is closest to 120 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 11 Homework, Page 356 45. Cathy Nyugen races on a bicycle with 13-inch radius wheels. When she is traveling at a speed of 44 ft/sec, how many revolutions per minute are her wheels making? 44 ft 60sec 12 in 1 rev 1 rpm 387.848 rpm sec 1min 1 ft 2 rad 13 in Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 12 Homework, Page 356 49. The captain of the tourist boat Julia follows a 038º course for 2 miles and then changes course to 047º for the next 4 miles. Draw a sketch of this trip. Endpoint Starting Point Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 13 Homework, Page 356 53. A simple pulley with the given radius r used to lift heavy objects is positioned 10 feet above ground level. Given the pulley rotates θº, determine the height to which the object is lifted. a. r = 4 in, θ = 720º s s r 4*720 * 16 50.265 in r 180 b. r = 2 ft, θ = 180º s s r 2*180 * 2 6.283 ft r 180 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 14 Homework, Page 356 57. If horse A is twice as far as horse B from the center of a merry-go-round, then horse A travels twice as fast as horse B. Justify your answer. s True, r s r r 2r s r 2r 1 1 2 1 2 2 1 since all points on a given radius have the same angular displacement, horse A will travel twice as far as horse B for the same angular displacement, thereby traveling twice as fast. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 15 Homework, Page 356 61. A bicycle with 26-inch diameter wheels is traveling at 10 mph. To the nearest whole number, how many revolutions does each wheel make per minute? a. 54 b. 129 c. 259 d. 406 e. 646 s r 10mi / hr 1rev 5280 ft 12in 1hr * * * * 129.283rpm 13in 2 rad 1mi 1 ft 60 min Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 16 Homework, Page 356 Find the difference in longitude between the given cities. 65. Minneapolis and Chicago 93 16 87 39 5 37 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 17 Homework, Page 356 Assume the cities have the same longitude and find the distance between them in nautical miles. 69. New Orleans and Minneapolis 44 59 2 29 57 152 15 15.03 60 15 2 60nm d *15.03 902 nautical miles 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 18 Homework, Page 356 73. Control tower A is 60 miles east of control tower B. At a certain time, an airplane bears 340º from control tower A and 037º from control tower B. Use a drawing to model the exact location of the airplane. y airplane x B A Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 19 4.2 Trigonometric Functions of Acute Angles Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review 1. Solve for x. x 2 3 2. Solve for x. 6 x 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 21 Quick Review 3. Convert 9.3 inches to feet. a 4. Solve for a. 0.45 20 36 5. Solve for b. 1.72 b Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 22 Quick Review Solutions 1. Solve for x. x 2 x 13 3 2. Solve for x. 6 x x3 3 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 23 Quick Review Solutions 3. Convert 9.3 inches to feet. 0.775 feet a 4. Solve for a. 0.45 9 20 36 5. Solve for b. 1.72 900 / 43 20.93 b Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 24 What you’ll learn about Right Triangle Trigonometry Two Famous Triangles Evaluating Trigonometric Functions with a Calculator Applications of Right Triangle Trigonometry … and why The many applications of right triangle trigonometry gave the subject its name. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 25 Leading Questions The functions y = secant x and y = cosecant x are reciprocal functions. Given the values of two primary trig functions, we can calculate the values of the others. Our left hand provides a key to the basic trig functions that is always with us. Given one angle and one side of a right triangle, we can find the other angle and sides. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 26 Standard Position An acute angle θ in standard position, with one ray along the positive x-axis and the other extending into the first quadrant. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 27 Trigonometric Functions Let be an acute angle in the right ABC. Then opp sine sin hyp adj cosine cos hyp opp tangent tan adj hyp cosecant csc opp hyp secant sec adj adj cotangent cot opp Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 28 Example Evaluating Trigonometric Functions of 45º Find the values of all six trigonometric functions for an angle of 45º. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 29 Example Evaluating Trigonometric Functions of 60º Find the values of all six trigonometric functions for an angle of 60º. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 30 Example Evaluating Trigonometric for General Triangles Find the values of all six trigonometric functions for the angle x in the triangle shown. 5 7 a Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley x Slide 4- 31 Trigonometric Functions of Five Common Angles sin cos 0 0 2 1 2 2 2 3 2 4 2 4 2 3 2 2 2 1 2 0 2 30 45 60 90 tan x sin cos tan x 0 0 0 0 1 0 0 30 1 2 2 2 3 2 3 2 2 2 1 2 3 3 3 3 1 3 D.N .E. 6 45 4 60 3 90 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 0 6 1 4 3 3 D.N .E. 2 Slide 4- 32 Trig Value Memory Aid Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 33 Common Calculator Errors When Evaluating Trig Functions Using the calculator in the wrong angle mode (degree/radians) Using the inverse trig keys to evaluate cot, sec, and csc Using function shorthand that the calculator does not recognize Not closing parentheses Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 34 Example Evaluating Trigonometric for General Triangles Find the exact value of the sine of 60º. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 35 Example Solving a Right Triangle A right triangle with a hypotenuse of 5 inches includes a 43 angle. Find the measures of the other two angles and the lengths of the other two sides. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 36 Example Solving a Word Problem Karen places her surveyor's telescope on the top of a tripod five feet above the ground. She measures an 8 elevation above the horizontal to the top of a tree that is 120 feet away. How tall is the tree? Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 37 Following Questions The circular functions get their name from the fact that we go around in circles trying to understand them. Angles are commonly measured counterclockwise from the initial side to the terminal side. Periods of functions are concerned with the frequency of their repetition. A unit circle has a diameter of one and is located wherever convenient. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 38 Homework Review Section 4.2 Page 366, Exercises: 1 – 73 (EOO) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 39 4.3 Trigonometry Extended: The Circular Functions Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Quick Review Give the value of the angle in degrees. 2 1. 3 2. 4 Use special triangles to evaluate. 3. cot 4 7 4. cos 6 5. Use a right triangle to find the other five trigonometric 4 functions of the acute angle given cos 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 41 Quick Review Solutions Give the value of the angle in degrees. 2 1. 120 3 2. 45 4 Use special triangles to evaluate. 3. cot 1 4 7 4. cos 3/2 6 5. Use a right triangle to find the other five trigonometric 4 functions of the acute angle given cos 5 sec 5 / 4, sin 3 / 5, csc 5 / 3, tan 3 / 4, cot 4 / 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 42 What you’ll learn about Trigonometric Functions of Any Angle Trigonometric Functions of Real Numbers Periodic Functions The 16-point unit circle … and why Extending trigonometric functions beyond triangle ratios opens up a new world of applications. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 43 Initial Side, Terminal Side Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 44 Positive Angle, Negative Angle Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 45 Coterminal Angles Two angles in an extended angle-measurement system can have the same initial side and the same terminal side, yet have different measures. Such angles are called coterminal angles. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 46 Example Finding Coterminal Angles Find a positive angle and a negative angle that are coterminal with 45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 47 Example Finding Coterminal Angles Find a positive angle and a negative angle that are coterminal with 6 . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 48 Example Evaluating Trig Functions Determined by a Point in Quadrant I Let be the acute angle in standard position whose terminal side contains the point (3,5). Find the six trigonometric functions of . Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 49 Trigonometric Functions of any Angle Let be any angle in standard position and let P( x, y ) be any point on the terminal side of the angle (except the origin). Let r denote the distance from P ( x, y ) to the origin, i.e., let r x y . Then 2 y sin r x cos r y tan ( x 0) x 2 r csc y r sec x x cot y Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley ( y 0) ( x 0) ( y 0) Slide 4- 50 Evaluating Trig Functions of a Nonquadrantal Angle θ 1. 2. 3. 4. 5. Draw the angle θ in standard position, being careful to place the terminal side in the correct quadrant. Without declaring a scale on either axis, label a point P (other than the origin) on the terminal side of θ. Draw a perpendicular segment from P to the x-axis, determining the reference triangle. If this triangle is one of the triangles whose ratios you know, label the sides accordingly. If it is not, then you will need to use your calculator. Use the sides of the triangle to determine the coordinates of point P, making them positive or negative according to the signs of x and y in that particular quadrant. Use the coordinates of point P and the definitions to determine the six trig functions. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 51 Example Evaluating More Trig Functions Find sin 210 without a calculator. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 52 Example Using one Trig Ratio to Find the Others Find sin and cos , given tan 4 / 3 and cos 0. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 53 Unit Circle The unit circle is a circle of radius 1 centered at the origin. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 54 Trigonometric Functions of Real Numbers Let t be any real number, and let P( x, y ) be the point corresponding to t when the number line is wrapped onto the unit circle as described above. Then sin t y cos t x y tan t ( x 0) x 1 csc t y 1 sec t x x cot t y ( y 0) ( x 0) ( y 0) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 55 Periodic Function A function y f (t ) is periodic if there is a positive number c such that f (t c) f (t ) for all values of t in the domain of f . The smallest such number c is called the period of the function. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 56 The 16-Point Unit Circle Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 4- 57