sin u 5 opp } cos u 5 adj } hyp opposite csc u 5 hyp } sec u 5 hyp } opp Using Inverse Trigonometric Functions

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1 13 Big Idea 1 CHAPTER SUMMARY BIG IDEAS Using Trigonometric Fnctions Algebra classzone.com Electronic Fnction Library For Yor Notebook hypotense acent osite sine cosine tangent sin 5 hyp cos 5 hyp tan 5 cosecant secant cotangent csc 5 hyp sec 5 hyp cot 5 Big Idea Using Inverse Trigonometric Fnctions Inverse trigonometric fnctions can be sed to solve trigonometric eqations. If 1 a 1, then the inverse sine of a is an angle, written sin 1 a 5, where sin 5 a and p p. If 1 a 1, then the inverse cosine of a is an angle, written cos 1 a 5, where cos 5 a and 0 π. If a is any real nmber, then the inverse tangent of a is an angle, written tan 1 a 5, where tan 5 a and p < < p sin cos1 Ï tan1 Ï Big Idea 3 Applying the Law of Sines and Law of Cosines Use the table below to help yo remember when to apply each law. If yo know this information... se this law... to find this information. angle-angle-side Law of sines remaining sides* angle-side-angle Law of sines remaining sides* side-side-angle Law of sines remaining side and one angle* side-angle-side Law of cosines remaining side and one angle* side-side-side Law of cosines two angles* * Find the remaining angle by sing the triangle sm theorem. Chapter Smmary 897 npe-1380.indd /16/05 1:34:16 PM

2 13 CHAPTER REVIEW REVIEW KEY VOCABULARY classzone.com Mlti-Langage Glossary Vocablary practice sine, p. 85 cosine, p. 85 tangent, p. 85 cosecant, p. 85 secant, p. 85 cotangent, p. 85 angle of elevation, p. 855 angle of depression, p. 855 initial side of an angle, p. 859 terminal side of an angle, p. 859 standard position of an angle, p. 859 coterminal angles, p. 860 radian, p. 860 sector, p. 861 central angle, p. 861 nit circle, p. 867 qadrantal angle, p. 867 reference angle, p. 868 inverse sine, p. 875 inverse cosine, p. 875 inverse tangent, p. 875 law of sines, p. 88 law of cosines, p. 889 VOCABULARY 1. WRITING Describe an angle in standard position.. Identify the relationship between the angles 58 and What is the name of a circle with center at the origin and radis 1 nit? 4. Copy and complete: If cos 5 a and 0 π, then the? of a eqals. 5. WRITING State the law of sines in words. REVIEW EXAMPLES AND Use the review examples and exercises below to check yor nderstanding of the concepts yo have learned in each lesson of Chapter Use Trigonometry with Right Triangles pp Evalate the six trigonometric fnctions of the angle. From the Pythagorean theorem, the length of the hypotense is Ï Ï sin 5 hyp csc 5 hyp cos 5 hyp sec 5 hyp tan cot EXAMPLES 1 and 3 on pp for Exs In n ABC, a 5 4, b 5 5, and C Evalate the six trigonometric fnctions of angle B. 7. HOT AIR BALLOON Yo are standing 50 meters from a hot air balloon that is preparing to take off. The angle of elevation to the top of the balloon is 88. Find the height of the balloon. 898 Chapter 13 Trigonometric Ratios and Fnctions npe-1380.indd /16/05 1:34:19 PM

3 classzone.com Chapter Review Practice 13. Define General Angles and Use Radian Measre pp Convert 1108 to radians and 7p radians to degrees p radians 7p 1808 radians 5 1 7p radians p radians p radians EXAMPLE 3 on p. 861 for Exs Convert the degree measre to radians or the radian measre to degrees p p Evalate Trigonometric Fnctions of Any Angle pp Evalate sec 08. The reference angle is The secant fnction is negative in Qadrant II, so yo can write: sec 08 5 sec EXAMPLE 4 on p. 869 for Exs. 15 Evalate the fnction withot sing a calclator.. tan csc (4058) 14. sin 13p sec 11p Evalate Inverse Trigonometric Fnctions pp Evalate tan 1 1 in both radians and degrees. When p < < p, or 908 < < 908, the angle whose tangent is 1 is: 5 tan p 4 or 5 tan EXAMPLES 1 and 4 on pp for Exs Evalate sin 1 (0.5) in both radians and degrees. 17. RAMP Yo se a foot ramp to load items into a van. If the floor of the van is 4 feet off the grond, what is the angle of elevation of the ramp? Chapter Review 899 npe-1380.indd /16/05 1:34:1 PM

4 CHAPTER REVIEW Apply the Law of Sines pp Solve n ABC with A 5 88, C 5 748, and b 5. Find angle B: B Use the law of sines to solve for a and c. a sin 88 5 sin 788 c sin sin 788 a 5 sin 88 ø 10.6 c 5 sin 748 ø 1.6 sin 788 sin 788 c For n ABC, B 5 788, a ø 10.6, and c ø 1.6. A b 5 88 c C 748 a B EXAMPLES 1,, 3 and 4 on pp for Exs Solve n ABC. (Hint: Some of the triangles may have no soltion and some may have two soltions.) 18. A 5 438, C 5 838, b B , b 5 5, c C 5 558, a 5 17, c B 5 608, C 5 738, b Apply the Law of Cosines pp Solve n ABC with A 5 668, b 5 16, and c 5 1. Use the law of cosines to find the length a. a 5 b 1 c bc cos A a (16)(1) cos 668 a ø 43.7 a ø 0.6 Now find angle B and angle C. sin B 16 5 sin sin B 5 16 sin 668 ø B 5 sin ø 45.8 C ø ø c For n ABC, B ø 45.8, C ø 68.88, and a ø 0.6. C b A c 5 1 B EXAMPLES 1 and on pp for Exs. 4 Solve n ABC.. a 5 19, b 5 11, c B 5 758, a 5 0, c a 5 30, b 5 35, c Chapter 13 Trigonometric Ratios and Fnctions npe-1380.indd /16/05 1:34:4 PM

5 13 CHAPTER TEST Evalate the six trigonometric fnctions of the angle Convert the degree measre to radians or the radian measre to degrees p p 3 Evalate the fnction withot sing a calclator. 8. tan sec(4808) 10. sin1 5p cos 11p 6 Evalate the expression in both radians and degrees withot sing a calclator.. cos tan 1 Ï sin 1 Ï 15. cos 1 Ï 3 Solve nabc. (Hint: Some of the triangles may have no soltion and some may have two soltions.) 16. A 5 478, C 5 38, c a 5 4, b 5, c B 5 638, a 5 11, b C , a 5 3, b a 5 4, b 5 30, c A 5 68, B 5 778, c 5 50 Find the area of nabc.. A 5 818, b 5 16, c = a 5 8, b 5 6, c a 5 5, b 5 4, c C , a 5 7, b a 5 16, b 5 33, c B 5 618, a 5, c SURVEYING To measre the width of a river, yo plant a stake at point A on one side of the riverbank, directly across from a tree stmp at point B on the other side of the riverbank. From point A, yo walk 80 meters along the riverbank to point C. Yo find the measre of angle C to be 398. What is the width w of the river? 9. CONSTRUCTION A crane has a 00 foot arm with a lower end that is 5 feet off the grond. The arm has to reach to the top of a bilding that is 160 feet high. At what angle shold the arm be set? 30. NAVIGATION A boat travels 40 miles de west before trning 08 and traveling an additional 5 miles. How far is the boat from its point of departre? Chapter Test 901

6 13 Standardized TEST PREPARATION Scoring Rbric Fll Credit soltion is complete and correct Partial Credit soltion is complete bt has errors, or soltion is withot error bt incomplete No Credit no soltion is given, or soltion makes no sense SHORT RESPONSE QUESTIONS P ROBLEM According to the ADA Accessibility Gidelines, the maximm slope of a wheelchair ramp is 1 :. What is the maximm angle of elevation of an acceptable ramp? What is the angle of elevation of the ramp shown? Does the ramp shown meet the gidelines? Explain yor reasoning. Not drawn to scale Below are sample soltions to the problem. Read each soltion and the comments in ble to see why the sample represents fll credit, partial credit, or no credit. SAMPLE 1: Fll credit soltion The diagram, calclations, and reasoning are correct. The diagram and calclations are correct and clearly explained. A ramp with the maximm acceptable slope is shown. From the diagram, tan 5 1, so 5 tan1 1 ø The maximm angle of elevation is abot The length of the ramp given in the problem is 30 feet. The rise of this ramp is 18 inches, or feet. So, tan and tan ø.868. The ramp s angle of elevation is abot.868. Ramp meeting ADA gidelines 30 ft ft The answer is correct. The ramp given in the problem has an angle of elevation less than the maximm acceptable angle, so the ramp meets the ADA gidelines. SAMPLE : Partial credit soltion The maximm acceptable angle is correctly calclated. The stdent forgot to se the same nits when calclating slope, so the rest of the answers are incorrect. The slope of an acceptable ramp can be at most 1 :, so tan 5 1 where is the maximm angle. Therefore, 5 tan 1 1 ø The maximm angle of elevation for a wheelchair ramp is abot The slope of the ramp given in the problem is 18 30, so tan Therefore, 5 tan 1 18 ø The ramp s angle of elevation is abot The angle of elevation exceeds the maximm acceptable angle of elevation, so the ramp does not meet the gidelines. 90 Chapter 13 Trigonometric Ratios and Fnctions npe-1390.indd 90 10/14/05 4:17:8 PM

7 SAMPLE 3: Partial credit soltion The answer is correct and clearly explained, bt only one of the three qestions is answered. The rise is 18 inches. The rn is 30 feet, x8 or 30 ft p in inches. 18 in. 1 ft 360 in. Yo know the lengths of the sides osite and acent to the angle, so se the tangent ratio. tan x x 5 tan ø.868 The angle of elevation of the ramp given in the problem is abot.868. SAMPLE 4: No credit soltion The stdent confsed slope and angle measre, and did not answer the final qestion. The maximm slope of an acceptable wheelchair ramp is 1 :, so the maximm angle of elevation is 1 ø The slope of the ramp shown in the problem is 18 in. : 30 ft, or 1 : 0, so the angle of elevation is PRACTICE Apply the Scoring Rbric Use the rbric on page 90 to score the soltion to the problem below as fll credit, partial credit, or no credit. Explain yor reasoning. PROBLEM Find the volme of the right trianglar prism shown cm 6 cm 1. To find the prism s volme, mltiply the three given measres: 30(6)(67) 5,060 cm 3. To find the area of the base of the prism, first find the side lengths b and h shown in the diagram. h cm sin b 30 30(sin 678) 5 b cos h 30 30(cos 678) 5 h b Base of prism 7.6 ø b 11.7 ø h The area of the base is 1 bh 5 1 ( 7.6) (11.7) ø cm. The volme of the prism is (area of base)(height) (6) cm 3. Standardized Test Preparation 903 npe-1390.indd /14/05 4:17:34 PM

8 13 Standardized TEST PRACTICE SHORT RESPONSE 1. Kepler s second law states that an imaginary line connecting the center of a planet and the center of the sn sweeps ot eqal areas in eqal time intervals. The diagram below shows the location of Mars in orbit over a ten day period. 4. A tennis player is practicing her serve. She aims for a can placed 57 feet from her in the service box. If she hits the ball when it is 9 feet in the air, what angle mst the path of the ball make with the grond in order for the ball to hit the can s base? Assme the ball travels along a straight line. Explain yor answer. sn AU AU Mars (position 1) Mars (position ) a. Approximate the area Mars swept ot dring this time by finding the area of the triangle formed by the sn and the starting and ending positions of Mars. Give the answer in sqare miles. (Hint: 1 AU is 1 astronomical nit, which is eqal to abot 93 million miles.) b. As a planet moves closer to the sn along its orbit, what does Kepler s law imply abot the planet s speed? Explain.. The wheel of a nicycle completes 6 fll revoltions. a. Throgh what angle, in degrees and radians, has the wheel rotated? b. If the diameter of the wheel is 4 inches, how far does the nicycle travel in 6 wheel revoltions? Explain. 3. The table lists several tools sed to make vertical transitions from one point to a higher point. It also lists the recommended range of angles of elevation for each tool. If a certain task reqires making a vertical transition with a slope of 3, which tool wold be appropriate 5 for the task? Explain yor reasoning. Tool Range of angles Ramp Fixed steps Ladder stairs Portable stairs Fixed ladder The diagrams below show the distances between atoms in a water molecle in liqid and ice forms. The measrements are given in picometers (pm), where 1 pm 5 10 m. Are the obtse angles in the diagrams the same? Explain. 96 pm 101 pm 96 pm 101 pm pm 165 pm Liqid 6. Yo are standing 30 feet from the base of a tree. The angle of elevation from yor eyes to the top of the tree is 708. If the height at eye level is 5 feet, what is the height of the tree to the nearest foot? Explain all of yor steps. 7. Yo kick a soccer ball with an initial velocity of 40 feet per second at an angle of 458. Yor teammate kicks a soccer ball at an angle of 58. With what initial velocity does yor teammate have to kick the ball in order for both of the soccer balls to travel the same horizontal distance? Explain yor reasoning. 8. Find the radis r (in inches) and the central angle (in radians) of a sector with an arc length of 10p inches and an area of 5p 3 3 sqare inches. Explain yor reasoning. Ice 904 Chapter 13 Trigonometric Ratios and Fnctions npe-1390.indd /14/05 4:17:38 PM

9 STATE TEST PRACTICE classzone.com MULTIPLE CHOICE 9. Which angle measre is shown in the diagram? A C p 4 radians B 3p 4 radians 5p 4 radians D 3p radians 10. What is the approximate vale of a in n ABC if A 5 858, B 5 78, and c cm? A 1.0 cm C 5.9 cm y x B 5.4 cm D 11.0 cm 11. What is the reference angle for 3008? A 308 B 608 C 08 D 408 GRIDDED ANSWER. If is an acte angle of a right triangle and sin 5 3, what is the vale of cos? What angle, in degrees, is eqivalent to 5p 6 radians? 14. What is the vale of sin in degrees? 15. What is the area of the triangle, to the nearest tenth of a sqare nit? Let (10, 4) be a point on the terminal side of an angle in standard position. What is the vale of sec? 17. What is the degree measre of angle A in n ABC if a 5 13, b 5 9, and c 5 7? Rond yor answer to the nearest degree. 7 EXTENDED RESPONSE 18. A boat ses 50 feet of rope to drop anchor in a lake. The angle that the rope makes with the bottom of the lake is 08. a. Find the depth of the water. b. The boat moves to deeper water bt still lets ot the same amont of rope when dring anchor. If the horizontal distance from the boat to the anchor is 37 feet, what angle does the rope make with the lake bottom? c. Describe how changes as the boat travels to deeper water with a constant length of anchor rope let ot. Assme the rope is always tat. 19. Yo are making a lampshade ot of fabric for the lamp shown. The pattern for the lampshade is shown in the diagram on the left. a. Use the smaller sector to write an eqation that relates and x. b. Use the larger sector to write an eqation that relates and x c. Solve the system of eqations from parts (a) and (b) to find x and. d. Use the formla for the area of a sector to find the amont of fabric (in sqare inches) that yo will se. 5π in. 14π in. x 10 in. Standardized Test Practice 905 npe-1390.indd /14/05 4:17:4 PM