Use Trigonometry with Right Triangles

Transcription

1 . a., a.4, 2A.2.A; G.5.D TEKS Use Trigonometr with Right Triangles Before Yo sed the Pthagorean theorem to find lengths. Now Yo will se trigonometric fnctions to find lengths. Wh? So o can measre distances indirectl, as in Eample 5. Ke Vocablar sine cosine tangent cosecant secant cotangent Consider a right triangle that has an acte angle (the Greek letter theta). The three sides of the triangle are the hpotense, the side opposite, and the side adjacent to. Ratios of a right triangle s side lengths are sed to define the si trigonometric fnctions: sine, cosine, tangent, cosecant, secant, and cotangent. These si fnctions are abbreviated sin, cos, tan, csc, sec, and cot, respectivel. KEY CONCEPT hpotense adjacent side For Yor Notebook opposite side Right Triangle Definitions of Trigonometric Fnctions Let be an acte angle of a right triangle. The si trigonometric fnctions of are defined as follows: sin 5} opposite cos 5} adjacent tan 5} opposite hpotense hpotense adjacent csc 5} hpotense sec 5} hpotense cot 5} adjacent opposite adjacent opposite The abbreviations opp, adj, and hp are often sed to represent the side lengths of the right triangle. Note that the ratios in the second row are reciprocals of the ratios in the first row: csc 5 } sin sec 5 } cos cot 5 } tan E XAMPLE Evalate trigonometric fnctions REVIEW GEOMETRY For help with the Pthagorean theorem, see p Evalate the si trigonometric fnctions of the angle. Soltion 5 From the Pthagorean theorem, the length of the hpotense is Ï } Ï } sin 5 } opp 5 } 2 hp cos 5 } adj 5 } 5 hp tan 5 } opp 5 } 2 adj 5 csc 5 } hp 5 opp 2 sec 5 } hp 5 adj 5 cot 5 } adj 5 } 5 opp 2 hpotense 2 52 Chapter Trigonometric Ratios and Fnctions

2 E XAMPLE 2 TAKS PRACTICE: Mltiple Choice If is an acte angle of a right triangle and sin 5 }, what is tan? A Ï} 55 } 55 B C Ï} 55 } D Soltion STEP Draw a right triangle with acte angle sch that the leg opposite has length and the hpotense has length. B the Pthagorean theorem, the length of the other leg is 5 Ï } Ï } 55. STEP 2 Find the vale of tan. tan 5 opp } adj 5 } Ï } 55 5 Ï} 55 } c The correct answer is A. A B C D GUIDED PRACTICE for Eamples and 2 Evalate the si trigonometric fnctions of the angle In a right triangle, is an acte angle and cos 5 7 } 0. What is sin? SPECIAL ANGLES The angles, 45, and 60 occr freqentl in trigonometr. Yo can se the trigonometric vales for these angles to find nknown side lengths in special right triangles. KEY CONCEPT For Yor Notebook Trigonometric Vales for Special Angles The table below gives the vales of the si trigonometric fnctions for the angles, 45, and 60. Yo can obtain these vales from the triangles shown. sin cos tan csc sec cot Ï Ï 2 2Ï Ï 45 Ï Ï Ï Ï 60 Ï }2 Ï 2Ï 2 Ï. Use Trigonometr with Right Triangles 5

3 E XAMPLE Find an nknown side length of a right triangle Find the vale of for the right triangle shown. Soltion Write an eqation sing a trigonometric fnction that involves the ratio of and. Solve the eqation for. cos 5} adj Write trigonometric eqation. hp Ï 5 } Sbstitte. 4Ï 5 Mltipl each side b. c The length of the side is 5 4Ï ø 6.9. at classzone.com SOLVING A TRIANGLE Finding all nknown side lengths and angle measres of a triangle is called solving the triangle. Solving right triangles that have acte angles other than, 45, and 60 ma reqire the se of a calclator. To find vales of the sine, cosine, and tangent fnctions on a calclator, se the kes,, and. Use these kes and the reciprocal ke for cosecant, secant, and cotangent. Be sre the calclator is set in degree mode. E XAMPLE 4 Use a calclator to solve a right triangle READING Throghot this chapter, a capital letter is sed to denote both an angle of a triangle and its measre. The same letter in lowercase is sed to denote the length of the side opposite that angle. Solve n ABC. Soltion A and B are complementar angles, so B tan 25 opp } adj sec 25 hp } adj Write trigonometric eqation. tan 25 a } 5 sec 25 c } 5 Sbstitte. 5(tan 2) 5 a 5} cos c Solve for the variable. A c 2 b 5 5 B a C 7.9 ø a 7.0 ø c Use a calclator. c So, B 5 62, a ø 7.9, and c ø 7.0. GUIDED PRACTICE for Eamples and 4 Solve n ABC sing the diagram at the right and the given measrements. 5. B 5 45, c A 5 2, b A 5 7, c B 5 60, a 5 7 A b c B a C 54 Chapter Trigonometric Ratios and Fnctions

4 E XAMPLE 5 Use indirect measrement CHOOSE FUNCTIONS The tangent fnction is sed to find the nknown distance becase it involves the ratio of and 2. GRAND CANYON While standing at Yavapai Point near the Grand Canon, o measre an angle of 90 between Powell Point and Widforss Point, as shown. Yo then walk to Powell Point and measre an angle of 76 between Yavapai Point and Widforss Point. The distance between Yavapai Point and Powell Point is abot 2 miles. How wide is the Grand Canon between Yavapai Point and Widforss Point? Soltion tan 765 Write trigonometric eqation. 2(tan 76) 5 Mltipl each side b 2..0 ø Use a calclator. c The width is abot.0 miles. Powell Point Widforss Point 76 2 mi Yavapai Point ANGLES OF SIGHT If o look at a point above o, sch as the top of a bilding, the angle that or line of sight makes with a line parallel to the grond is called the angle of elevation. At the top of the bilding, the angle between a line parallel to the grond and or line of sight is called the angle of depression. These two angles have the same measre. o angle of depression angle of elevation E XAMPLE 6 Use an angle of elevation PARASAILING A parasailer is attached to a boat with a rope 00 feet long. The angle of elevation from the boat to the parasailer is 4. Estimate the parasailer s height above the boat. Soltion STEP Draw a diagram that represents the sitation. STEP 2 Write and solve an eqation to find the height h. 00 ft h sin 45 h 00 Write trigonometric eqation. 4 00(sin 4) 5 h Mltipl each side b ø h Use a calclator. c The height of the parasailer above the boat is abot 22 feet. GUIDED PRACTICE for Eamples 5 and 6 9. GRAND CANYON In Eample 5, find the distance between Powell Point and Widforss Point. 0. WHAT IF? In Eample 6, estimate the height of the parasailer above the boat if the angle of elevation is.. Use Trigonometr with Right Triangles 55

5 . EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5,, and 5 TAKS PRACTICE AND REASONING Es. 5, 20,, 6,, and 9 5 MULTIPLE REPRESENTATIONS E. 4. VOCABULARY What is an angle of elevation? 2. WRITING Eplain what it means to solve a right triangle. EXAMPLE on p. 52 for Es. EVALUATING FUNCTIONS Evalate the si trigonometric fnctions of the angle EXAMPLE 2 on p. 5 for Es. 9 6 FINDING VALUES Let be an acte angle of a right triangle. Find the vales of the other five trigonometric fnctions of. 9. sin 5 5 } 6 0. cos 5 5 }. tan csc 5 0 } 7. sec 5 2 } 5 4. cot 5 6 } 5. TAKS REASONING In a right triangle, is an acte angle and cos 5 } 4. 9 What is the vale of tan? A 4Ï} 65 } 65 B Ï} 65 } 9 C Ï} 65 } 4 D 9 } 4 6. ERROR ANALYSIS Describe and correct the error in finding csc, given that is an acte angle of a right triangle and cos 5 } 7. csc 5 } cos 5 } 7 EXAMPLE on p. 54 for Es FINDING SIDE LENGTHS Find the eact vales of and TAKS REASONING In a triangle, the longer leg has a length of 5. What is the length of the shorter leg? A 5Ï B 5Ï C 0Ï D 5Ï 56 Chapter Trigonometric Ratios and Fnctions

6 EXAMPLE 4 on p. 54 for Es. 2 2 SOLVING TRIANGLES Solve n ABC sing the diagram and the given measrements. 2. A 5 5, c B 5 5, a 5 2 A 2. B 5, c A 5 67, b 5 7 c b 25. B 5 75, a A 5 49, c A 5 64, b B 5 24, c 5 0. C a B 29. CHALLENGE A procedre for approimating π based on the work of Archimedes is to inscribe a reglar heagon in a circle. a. Use the diagram at the right to solve for. What is the perimeter of the heagon? b. Show that a reglar n-sided polgon inscribed in a circle of radis has a perimeter of 2n p sin 0 } n 2. c. Use the reslt from part (b) to find an epression in terms of n that approimates π. Then evalate the epression when n PROBLEM SOLVING EXAMPLES 5 and 6 on p. 55 for Es. 0 5 In Eercises 0 and, se the information in the diagram to solve the problem. 0. TREE HEIGHT A tree casts the shadow shown. What is the height of the tree?. GRAND PIANO Find the length of the prop holding open the piano. 25 ft 2. RAILWAY The Falls Incline Railwa at Niagara Falls has an angle of elevation of 6. The railwa etends a horizontal distance of abot feet. Find the height and length of the railwa.. TAKS REASONING A sbmersible traveling at a depth of 250 feet dives at an angle of 5 with respect to a line parallel to the water s srface. It travels a horizontal distance of 500 feet dring the dive. What is the depth of the sbmersible after the dive? Eplain how the angle of the dive affects the final depth. 4. MULTIPLE REPRESENTATIONS Yo are climbing Mont Massive in Colorado. Yo are at an altitde of,200 feet. Yo measre the angle of elevation to a ridge above o to be The distance (along the face of the montain) between o and the ridge is 65 feet. a. Drawing a Diagram Draw a diagram that represents this sitation. b. Writing an Eqation Write and solve an eqation to find the altitde of the ridge.. Use Trigonometr with Right Triangles 57

7 5. TROPIC OF CANCER The Tropic of Cancer is the circle of latitde farthest north of the eqator where the sn can appear directl overhead. It lies 2.5 north of the eqator, as shown. a. Find the circmference of the Tropic of Cancer sing 960 miles as Earth s approimate radis. b. What is the distance between two points on the Tropic of Cancer that lie directl across from each other? Tropic of Cancer eqator North Pole Soth Pole 6. TAKS REASONING A passenger in an airplane sees two towns directl to the left of the plane ,000 ft d a. What is the distance d from the airplane to the first town? b. What is the horizontal distance from the airplane to the first town? c. What is the distance between the two towns? Eplain the process o sed to find or answer. 7. CHALLENGE Yo measre the angle of elevation from the grond to the top of a bilding as 2. When o move 50 meters closer to the bilding, the angle of elevation is 5. How high is the bilding? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 4.; TAKS Workbook. TAKS PRACTICE The height h (in feet) of a horseshoe tossed dring a game of horseshoes is h 526t 2 0t 2 where t is the time (in seconds). Abot how long is the horseshoe in the air? TAKS Obj. 5 A.4 sec C 2.9 sec B.9 sec D.6 sec REVIEW Lesson 9.; TAKS Workbook 9. TAKS PRACTICE Rectangle KLMN has diagonals that intersect at point P. What are the coordinates of point L? TAKS Obj. 7 K L F (2, 5) P (, ) G (2, 4) H (5, 5) J (5, 4) N (2, 2) M 5 EXTRA PRACTICE for Lesson., p. 022 ONLINE QUIZ at classzone.com