Applying Trigonometric Functions. ENTERTAINMENT The circus has arrived and the roustabouts must put

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1 5-4 OJETIVE Use trigonometry to find te measures of te sides of rigt triangles. pplying Trigonometric Functions ENTERTINMENT Te circus as arrived and te roustabouts must put up te main tent in a field near town. tab is located on te side of te tent 40 feet above te ground. rope is tied to te tent at tis point and ten te rope is placed around a stake on te ground. If te angle tat te rope makes wit te level ground is 50 15, ow long is te rope? Wat is te distance between te bottom of te tent and te stake? Tis problem will be solved in Example 2. p plic atio n Trigonometric functions can be used to solve problems involving rigt triangles. Te most common functions used are te sine, cosine, and tangent. Examples 1 If P 35 and r 14, find q. From te art at te rigt, you know te measures of an angle and te ypotenuse. You want to know te measure of te side adjacent to te given angle. Te cosine function relates te side adjacent to te angle and te ypotenuse. P cos P q r cos s ide adjacent ypotenuse q cos 35 Substitute 35 for P and 14 for r cos 35 q Multiply eac side by q r 14 R p Q q Use a calculator. Terefore, q is about p plic atio n 2 ENTERTINMENT Refer to te application above. a. If te angle tat te rope makes wit te level ground is 52 15, ow long is te rope? b. Wat is te distance between te bottom of te tent and te stake? a. You know te measures of an angle and te side opposite te angle. To find te r 40 ft lengt of te rope, you need to know te measure of te ypotenuse. In tis case, d use te sine function. (continued on te next page) Lesson 5-4 pplying Trigonometric Functions 299

2 sin side opposite sin r ypotenuse r sin Multiply eac side by r. 40 r Divide eac side by sin sin r Use a calculator. Te rope is about 50.6 feet long. b. To find te distance between te bottom of te tent and te stake, you need to know te lengt of te side adjacent to te known angle. Use te tangent function. tan tan s ide opposite d side adjacent d tan Multiply eac side by d. 40 d Divide eac side by tan tan d Use a calculator. Te distance between te bottom of te tent and te stake is about 31.0 feet. You can use rigt triangle trigonometry to solve problems involving oter geometric figures. Example 3 GEOMETRY regular pentagon is inscribed in a circle wit diameter 8.34 centimeters. Te apotem of a regular polygon is te measure of a line segment from te center of te polygon to te midpoint of one of its sides. Find te apotem of te pentagon. First, draw a diagram. If te diameter of te circle is 8.34 centimeters, te radius is or 4.17 centimeters. Te measure of is or 36. cos 36 4ạ 17 cos s ide adjacent ypotenuse cos 36 a Multiply eac side by cm a a Use a calculator. Te apotem is about 3.37 centimeters. Horizontal ngle of Depression ngle of Elevation Horizontal Tere are many oter applications tat require trigonometric solutions. For example, surveyors use special instruments to find te measures of angles of elevation and angles of depression. n angle of elevation is te angle between a orizontal line and te line of sigt from an observer to an object at a iger level. n angle of depression is te angle between a orizontal line and te line of sigt from te observer to an object at a lower level. Te angle of elevation and te angle of depression are equal in measure because tey are alternate interior angles. 300 apter 5 Te Trigonometric Functions

3 Example 4 p plic atio n SURVEYING On May 18, 1980, Mount Saint Helens, a volcano in Wasington, erupted wit suc force tat te top of te mountain was blown off. To determine te new eigt at te summit of Mount Saint Helens, a surveyor measured te angle of elevation to te top of te volcano to be Te surveyor ten moved 1000 feet closer to te volcano and measured te angle of elevation to be Determine te new eigt of Mount Saint Helens. Draw a diagram to model te situation. Let represent te eigt of te volcano and x represent te distance from te surveyor s second position to te center of te base of te volcano. Write two equations involving te tangent function. tan x (1000 x)tan x tan x x tan ft x Terefore, (1000 x)tan x tan Solve tis equation for x. (1000 x)tan 37 46x tan tan x tan 37 46x tan tan 37 46x tan x tan tan 37 46x(tan tan 37 46) 1000 tan x tan tan x Use a calculator. Use tis value for x and te equation x tan to find te eigt of te volcano. x tan tan Use a calculator. Te new eigt of Mount Saint Helens is about 8341 feet. HEK FOR U NDERSTNDING ommunicating Matematics Read and study te lesson to answer eac question. 1. State wic trigonometric function you would use to solve eac problem. a. If S 42 and ST 8, find RS. b. If T 55 and RT 5, find RS. c. If S 27 and TR 7, find TS. 2. Write a problem tat could be solved using te tangent function. S T R Lesson 5-4 pplying Trigonometric Functions 301

4 Guided Practice 3. Name te angle of elevation and te angle of depression in te figure at te rigt. ompare te measures of tese angles. Explain. 4. Describe a way to use trigonometry to determine te eigt of te building were you live. Solve eac problem. Round to te nearest tent. 5. If b 13 and 76, find a. 6. If 26 and b 18, find c. 7. If and c 13, find a. 8. Geometry Eac base angle of an isosceles triangle measures Eac of te congruent sides is 10 centimeters long. a. Find te altitude of te triangle. b. Wat is te lengt of te base? c. Find te area of te triangle. 9. oating Te Ponce de Leon ligtouse in St. ugustine, Florida, is te second tallest brick tower in te United States. It was built in 1887 and rises 175 feet above sea level. How far from te sore is a motorboat if te angle of depression from te top of te ligtouse is 13 15? b a c D Practice E XERISES Solve eac problem. Round to te nearest tent. 10. If 37 and b 6, find a. 11. If c 16 and = 67, find a. 12. If 62 and c 24, find b. 13. If 29 and a 4.6, find c. 14. If a 17.3 and 77, find c. 15. If b 33.2 and 61, find a. 16. If and b 10, find a. 17. If and c 13.7, find a. 18. If a 22.3 and 47 18, find c. 19. Find, n, m, and p. Round to te nearest tent. 20. Geometry Te apotem of a regular pentagon is 10.8 centimeters. a. Find te radius of te circumscribed circle. b. Wat is te lengt of a side of te pentagon? c. Find te perimeter of te pentagon. 21. Geometry Eac base angle of an isosceles triangle measures Te base is 14.6 meters long. a. Find te lengt of a leg of te triangle. b. Find te altitude of te triangle. c. Wat is te area of te triangle? c a Exercises p m n Exercise 19 b 302 apter 5 Te Trigonometric Functions

5 22. Geometry regular exagon is inscribed in a circle wit diameter 6.4 centimeters. a. Wat is te apotem of te exagon? b. Find te lengt of a side of te exagon. c. Find te perimeter of te exagon. d. Te area of a regular polygon equals one alf times te perimeter of te polygon times te apotem. Find te area of te polygon. pplications and Problem Solving 23. Engineering Te escalator at St. Petersburg Metro in Russia as a vertical rise of feet. If te angle of elevation of te escalator is , find te lengt of te escalator. p plic atio n 24. ritical Tinking Write a formula for te volume of te regular pyramid at te rigt in terms of and s te lengt of eac side of te base. α 25. Fire Figting Te longest truck-mounted ladder used by te Dallas Fire Department is 108 feet long and consists of four ydraulic sections. Gerald Travis, aerial expert for te department, indicates tat te optimum operating angle of tis ladder is 60. Te fire figters find tey need to reac te roof of an 84-foot burning building. ssume te ladder is mounted 8 feet above te ground. a. Draw a labeled diagram of te situation. b. How far from te building sould te base of te ladder be placed to acieve te optimum operating angle? c. How far sould te ladder be extended to reac te roof? s 26. viation Wen a 757 passenger jet begins its descent to te Ronald Reagan International irport in Wasington, D.., it is 3900 feet from te ground. Its angle of descent is 6. a. Wat is te plane s ground distance to te airport? b. How far must te plane fly to reac te runway? 3900 ft 6 Ground Distance Distance Traveled irport 27. oat Safety Te ape Hatteras ligtouse on te Nort arolina coast was built in 1870 and rises 208 feet above sea level. From te top of te ligtouse, te ligtouse keeper observes a yact and a barge along te same line of sigt. Te angle of depression for te yact is 20, and te angle of depression for te barge is For safety purposes, te keeper tinks tat te two sea vessels sould be at least 300 feet apart. If tey are less tan 300 feet, se plans to sound te orn. How far apart are tese vessels? Does te keeper ave to sound te orn? 28. ritical Tinking Derive two formulas for te lengt of te altitude a of te triangle sown at te rigt, given tat b, s, and are known. Justify eac of te steps you take in your reasoning. E s G b a s F Lesson 5-4 pplying Trigonometric Functions 303

6 29. Recreation Latasa and Markisa are flying kites on a windy spring day. Latasa as released 250 feet of string, and Markisa as released 225 feet of string. Te angle tat Latasa s kite string makes wit te orizontal is 35. Te angle tat Markisa s kite string makes wit te orizontal is 42. Wic kite is iger and by ow muc? 30. rcitecture flagpole 40 feet ig stands on top of te Wentwort uilding. From a point in front of ailey s Drugstore, te angle of elevation for te top of te pole is 54 54, and te angle of elevation for te bottom of te pole is How ig is te building? Mixed Review 31. Find te values of te six trigonometric functions for a 120 angle using te unit circle. (Lesson 5-3) 32. Find te sine, cosine, and tangent ratios for P. (Lesson 5-2) 33. Write as a decimal to te nearest tousandt. (Lesson 5-1) 34. Grap y x 2. (Lesson 3-3) 35. onsumerism Kareem and Erin went sopping for scool supplies. Kareem bougt 3 notebooks and 2 packages of pencils for $5.80. Erin bougt 4 notebooks and 1 package of pencils for $6.20. Wat is te cost of one notebook? Wat is te cost of one package of pencils? (Lesson 2-1) 36. ST/T Practice n automobile travels m miles in ours. t tis rate, ow far will it travel in x ours? m x m x m D m E m x x P in Q 2 in. R MID-HPTER QUIZ 1. ange to degrees, minutes, and seconds. (Lesson 5-1) 2. If a 400 angle is in standard position, determine a coterminal angle tat is between 0 and 360. State te quadrant in wic te terminal side lies. (Lesson 5-1) 3. Find te six trigonometric functions for G. (Lesson 5-2) G I 12 m 10 m H 4. Find te values of te six trigonometric functions for angle in standard position if a point wit coordinates (2, 5) lies on its terminal side. (Lesson 5-3) 5. National Landmarks Suppose te angle of elevation of te sun is Find te lengt of te sadow made by te Wasington Monument, wic is 550 feet tall. (Lesson 5-4) 304 apter 5 Te Trigonometric Functions Extra Practice See p. 34.