6.2 TRIGONOMETRY OF RIGHT TRIANGLES

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1 8 CHAPTER 6 Trigonometric Functions: Rigt Triangle Approac 6. TRIGONOMETRY OF RIGHT TRIANGLES Trigonometric Ratios Special Triangles; Calculators Applications of Trigonometry of Rigt Triangles In tis section we study certain ratios of te sides of rigt triangles, called trigonometric ratios, and give several applications. Trigonometric Ratios Consider a rigt triangle wit u as one of its acute angles. Te trigonometric ratios are defined as follows (see Figure ). ypotenuse adjacent FIGURE opposite THE TRIGONOMETRIC RATIOS sin u opposite ypotenuse csc u ypotenuse opposite cos u adjacent ypotenuse sec u ypotenuse adjacent tan u opposite adjacent cot u adjacent opposite Te symbols we use for tese ratios are abbreviations for teir full names: sine, cosine, tangent, cosecant, secant, cotangent. Since any two rigt triangles wit angle u are similar, tese ratios are te same, regardless of te size of te triangle; te trigonometric ratios depend only on te angle u (see Figure ) FIGURE ß = ß = 0 = 0 œ FIGURE EXAMPLE Finding Trigonometric Ratios Find te si trigonometric ratios of te angle u in Figure. SOLUTION By te definition of trigonometric ratios, we get Now Try Eercise sin u csc u cos u! sec u! EXAMPLE Finding Trigonometric Ratios tan u! cot u! If cos a, sketc a rigt triangle wit acute angle a, and find te oter five trigonometric ratios of a. Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

2 SECTION 6. Trigonometry of Rigt Triangles 8 œ 7 SOLUTION Since cos a is defined as te ratio of te adjacent side to te ypotenuse, we sketc a triangle wit ypotenuse of lengt and a side of lengt adjacent to a. If te opposite side is, ten by te Pytagorean Teorem, or 7, so!7. We ten use te triangle in Figure to find te ratios. sin a!7 cos a tan a!7 å csc a!7 sec a cot a!7 FIGURE Now Try Eercise Special Triangles; Calculators Tere are special trigonometric ratios tat can be calculated from certain triangles (wic we call special triangles). We can also use a calculator to find trigonometric ratios. HIPPARCHUS (circa 0 B.C.) is considered te founder of trigonometry. He constructed tables for a function closely related to te modern sine function and evaluated for angles at alf-degree intervals. Tese are considered te first trigonometric tables. He used is tables mainly to calculate te pats of te planets troug te eavens. Special Ratios Certain rigt triangles ave ratios tat can be calculated easily from te Pytagorean Teorem. Since tey are used frequently, we mention tem ere. Te first triangle is obtained by drawing a diagonal in a square of side (see Figure ). By te Pytagorean Teorem tis diagonal as lengt!. Te resulting triangle as angles,, and 90 (or p/, p/, and p/). To get te second triangle, we start wit an equilateral triangle ABC of side and draw te perpendicular bisector DB of te base, as in Figure 6. By te Pytagorean Teorem te lengt of DB is!. Since DB bisects angle ABC, we obtain a triangle wit angles 0, 60, and 90 (or p/6, p/, and p/). œ * FIGURE * A 60* FIGURE 6 B 0* œ D C We can now use te special triangles in Figures and 6 to calculate te trigonometric ratios for angles wit measures 0,, and 60 (or p/6, p/, and p/). Tese are listed in te table below. SPECIAL VALUES OF THE TRIGONOMETRIC FUNCTIONS Te following values of te trigonometric functions are obtained from te special triangles. œ * u in degrees u in radians sin u cos u tan u csc u sec u cot u p 6 p p!!!!!!!!!!! p 0 0! * 60* 0* œ Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

3 8 CHAPTER 6 Trigonometric Functions: Rigt Triangle Approac For an eplanation of numerical metods, see te margin note on page. It s useful to remember tese special trigonometric ratios because tey occur often. Of course, tey can be recalled easily if we remember te triangles from wic tey are obtained. Using a Calculator To find te values of te trigonometric ratios for oter angles, we use a calculator. Matematical metods (called numerical metods) used in finding te trigonometric ratios are programmed directly into scientific calculators. For instance, wen te SIN key is pressed, te calculator computes an approimation to te value of te sine of te given angle. Calculators give te values of sine, cosine, and tangent; te oter ratios can be easily calculated from tese by using te following reciprocal relations: csc t sin t sec t cos t cot t tan t You sould ceck tat tese relations follow immediately from te definitions of te trigonometric ratios. We follow te convention tat wen we write sin t, we mean te sine of te angle wose radian measure is t. For instance, sin means te sine of te angle wose radian measure is. Wen using a calculator to find an approimate value for tis number, set your calculator to radian mode; you will find tat sin If you want to find te sine of te angle wose measure is, set your calculator to degree mode; you will find tat sin EXAMPLE Using a Calculator Using a calculator, find te following. (a) tan 0 (b) cos 0 (c) cot (d) csc 80 SOLUTION Making sure our calculator is set in degree mode and rounding te results to si decimal places, we get te following: (a) tan (b) cos (c) cot.0078 (d) csc 80 tan sin Now Try Eercise Applications of Trigonometry of Rigt Triangles A triangle as si parts: tree angles and tree sides. To solve a triangle means to determine all of its parts from te information known about te triangle, tat is, to determine te lengts of te tree sides and te measures of te tree angles. Hulton Arcive/Moviepi/Getty Images DISCOVERY PROJECT Similarity Similarity of triangles is te basic concept underlying te definition of te trigonometric functions. Te ratios of te sides of a triangle are te same as te corresponding ratios in any similar triangle. But te concept of similarity of figures applies to all sapes, not just triangles. In tis project we eplore ow areas and volumes of similar figures are related. Tese relationsips allow us to determine weter an ape te size of King Kong (tat is, an ape similar to, but muc larger tan, a real ape) can actually eist. You can find te project at Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

SECTION 6. Trigonometry of Rigt Triangles 8 EXAMPLE Solving a Rigt Triangle B Solve triangle ABC, sown in Figure 7. A FIGURE 7 0* b a C SOLUTION It s clear tat B 60.

Evaluate FIGURE 8 a r sin u, b r cos u r b ARISTARCHUS OF SAMOS (0 0 B.C.) was a famous Greek scientist, musician, astronomer, and geometer.

4 SECTION 6. Trigonometry of Rigt Triangles 8 EXAMPLE Solving a Rigt Triangle B Solve triangle ABC, sown in Figure 7. A FIGURE 7 0* b a C SOLUTION It s clear tat B 60. From Figure 7 we ave sin 0 a Definition of sine a sin 0 Multiply by A B 6 Evaluate Also from Figure 7 we ave cos 0 b Definition of cosine b cos 0 Multiply by a! b 6! Evaluate FIGURE 8 a r sin u, b r cos u r b ARISTARCHUS OF SAMOS (0 0 B.C.) was a famous Greek scientist, musician, astronomer, and geometer. He observed tat te angle between te sun and moon can be measured directly (see te figure below). In is book On te Sizes and Distances of te Sun and te Moon e estimated te distance to te sun by observing tat wen te moon is eactly alf full, te triangle formed by te sun, te moon, and te eart as a rigt angle at te moon. His metod was similar to te one described in Eercise 67 in tis section. Aristarcus was te first to advance te teory tat te eart and planets move around te sun, an idea tat did not gain full acceptance until after te time of Copernicus, 800 years later. For tis reason Aristarcus is often called te Copernicus of antiquity. a Now Try Eercise 7 Figure 8 sows tat if we know te ypotenuse r and an acute angle u in a rigt triangle, ten te legs a and b are given by a r sin u and b r cos u Te ability to solve rigt triangles by using te trigonometric ratios is fundamental to many problems in navigation, surveying, astronomy, and te measurement of distances. Te applications we consider in tis section always involve rigt triangles, but as we will see in te net tree sections, trigonometry is also useful in solving triangles tat are not rigt triangles. To discuss te net eamples, we need some terminology. If an observer is looking at an object, ten te line from te eye of te observer to te object is called te line of sigt (Figure 9). If te object being observed is above te orizontal, ten te angle between te line of sigt and te orizontal is called te angle of elevation. If te object is below te orizontal, ten te angle between te line of sigt and te orizontal is called te angle of depression. In many of te eamples and eercises in tis capter, angles of elevation and depression will be given for a ypotetical observer at ground level. If te line of sigt follows a pysical object, suc as an inclined plane or a illside, we use te term angle of inclination. Line of sigt Angle of elevation Horizontal Angle of depression Horizontal Line of sigt FIGURE 9 Te net eample gives an important application of trigonometry to te problem of measurement: We measure te eigt of a tall tree witout aving to climb it! Altoug te eample is simple, te result is fundamental to understanding ow te trigonometric ratios are applied to suc problems. Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

86 CHAPTER 6 Trigonometric Functions: Rigt Triangle Approac THALES OF MILETUS (circa 6 7 B.C.) is te legendary founder of Greek geometry.

Using properties of similar triangles, e argued tat te ratio of te eigt of te column to te eigt of is staff was equal to te ratio of te lengt s of te column s sadow to te lengt s of te staff s sadow:

According to legend, Tales used a similar metod to find te eigt of te Great Pyramid in Egypt, a feat tat impressed Egypt s king.

Te principle Tales used, te fact tat ratios of corresponding sides of similar triangles are equal, is te foundation of te subject of trigonometry.

5 86 CHAPTER 6 Trigonometric Functions: Rigt Triangle Approac THALES OF MILETUS (circa 6 7 B.C.) is te legendary founder of Greek geometry. It is said tat e calculated te eigt of a Greek column by comparing te lengt of te sadow of is staff wit tat of te column. Using properties of similar triangles, e argued tat te ratio of te eigt of te column to te eigt of is staff was equal to te ratio of te lengt s of te column s sadow to te lengt s of te staff s sadow: r s sr Since tree of tese quantities are known, Tales was able to calculate te eigt of te column. According to legend, Tales used a similar metod to find te eigt of te Great Pyramid in Egypt, a feat tat impressed Egypt s king. Plutarc wrote tat altoug e [te king of Egypt] admired you [Tales] for oter tings, yet e particularly liked te manner by wic you measured te eigt of te pyramid witout any trouble or instrument. Te principle Tales used, te fact tat ratios of corresponding sides of similar triangles are equal, is te foundation of te subject of trigonometry. FIGURE * 7* 00 ft k EXAMPLE Finding te Heigt of a Tree A giant redwood tree casts a sadow ft long. Find te eigt of te tree if te angle of elevation of te sun is.7. SOLUTION Let te eigt of te tree be. From Figure 0 we see tat tan.7 Definition of tangent tan.7 Multiply by Terefore te eigt of te tree is about 6 ft. Now Try Eercise FIGURE 0.7* ft Use a calculator EXAMPLE 6 A Problem Involving Rigt Triangles From a point on te ground 00 ft from te base of a building, an observer finds tat te angle of elevation to te top of te building is and tat te angle of elevation to te top of a flagpole atop te building is 7. Find te eigt of te building and te lengt of te flagpole. SOLUTION Figure illustrates te situation. Te eigt of te building is found in te same way tat we found te eigt of te tree in Eample. tan 00 Definition of tangent 00 tan Multiply by Use a calculator Te eigt of te building is approimately ft. To find te lengt of te flagpole, let s first find te eigt from te ground to te top of te pole. k tan 7 00 Definition of tangent k 00 tan 7 Multiply by Use a calculator To find te lengt of te flagpole, we subtract from k. So te lengt of te pole is approimately ft. Now Try Eercise 6 Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

6 SECTION 6. Trigonometry of Rigt Triangles EXERCISES CONCEPTS. A rigt triangle wit an angle u is sown in te figure. 9 0 Trigonometric Ratios Find (a) sin a and cos b, (b) tan a and cot b, and (c) sec a and csc b. 9. å 0. å 7 (a) Label te opposite and adjacent sides of u and te ypotenuse of te triangle. (b) Te trigonometric functions of te angle u are defined as follows: sin u cos u tan u (c) Te trigonometric ratios do not depend on te size of te triangle. Tis is because all rigt triangles wit te same acute angle u are.. Te reciprocal identities state tat csc u sec u cot u Using a Calculator Use a calculator to evaluate te epression. Round your answer to five decimal places.. (a) sin (b) cot. (a) cos 7 (b) csc 8. (a) sec (b) tan. (a) csc 0 (b) sin 6 0 Finding an Unknown Side Find te side labeled. In Eercises 7 and 8 state your answer rounded to five decimal places * 60* * * SKILLS 8 Trigonometric Ratios Find te eact values of te si trigonometric ratios of te angle u in te triangle * * Trigonometric Ratios Epress and y in terms of trigonometric ratios of u.. 8 y. y Trigonometric Ratios Sketc a triangle tat as acute angle u, and find te oter five trigonometric ratios of u.. tan u 6. cos u. cot u 7 6. tan u! 7. csc u 6 8. cot u Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

7 88 CHAPTER 6 Trigonometric Functions: Rigt Triangle Approac 9 6 Evaluating an Epression Evaluate te epression witout using a calculator. 9. sin p 6 cos p Finding an Unknown Side Find rounded to one decimal place sin 0 csc 0. sin 0 cos 60 sin 60 cos 0. sin 60 cos 60. cos 0 sin 0. a sin p cos p sin p cos p b * 0*. a cos p sin p 6 b 6. a sin p tan p 6 csc p b 7 Solving a Rigt Triangle Solve te rigt triangle * 0* * * 6 7* * 6* 0*. Trigonometric Ratios Epress te lengt in terms of te trigonometric ratios of u... π π π 8 π SKILLS Plus. Using a Ruler to Estimate Trigonometric Ratios Use a ruler to carefully measure te sides of te triangle, and ten use your measurements to estimate te si trigonometric ratios of u.. Trigonometric Ratios Epress te lengts a, b, c, and d in te figure in terms of te trigonometric ratios of u. c a b d 6. Using a Protractor to Estimate Trigonometric Ratios Using a protractor, sketc a rigt triangle tat as te acute angle 0. Measure te sides carefully, and use your results to estimate te si trigonometric ratios of 0. Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

Using tis information, find te eigt of te Empire State Building.. Gateway Arc A plane is flying witin sigt of te Gateway Arc in St. Louis, Missouri, at an elevation of,000 ft.

8 SECTION 6. Trigonometry of Rigt Triangles 89 APPLICATIONS. Heigt of a Building Te angle of elevation to te top of te Empire State Building in New York is found to be from te ground at a distance of mi from te base of te building. Using tis information, find te eigt of te Empire State Building.. Gateway Arc A plane is flying witin sigt of te Gateway Arc in St. Louis, Missouri, at an elevation of,000 ft. Te pilot would like to estimate er distance from te Gateway Arc. Se finds tat te angle of depression to a point on te ground below te arc is. (a) Wat is te distance between te plane and te arc? (b) Wat is te distance between a point on te ground directly below te plane and te arc?. Deviation of a Laser Beam A laser beam is to be directed toward te center of te moon, but te beam strays 0. from its intended pat. (a) How far as te beam diverged from its assigned target wen it reaces te moon? (Te distance from te eart to te moon is 0,000 mi.) (b) Te radius of te moon is about 000 mi. Will te beam strike te moon? 6. Distance at Sea From te top of a 00-ft ligtouse, te angle of depression to a sip in te ocean is. How far is te sip from te base of te ligtouse? 7. Leaning Ladder A 0-ft ladder leans against a building so tat te angle between te ground and te ladder is 7. How ig does te ladder reac on te building? 8. Heigt of a Tower A 600-ft guy wire is attaced to te top of a communications tower. If te wire makes an angle of 6 wit te ground, ow tall is te communications tower? 9. Elevation of a Kite A man is lying on te beac, flying a kite. He olds te end of te kite string at ground level and estimates te angle of elevation of te kite to be 0. If te string is 0 ft long, ow ig is te kite above te ground? 60. Determining a Distance A woman standing on a ill sees a flagpole tat se knows is 60 ft tall. Te angle of depression to te bottom of te pole is, and te angle of elevation to te top of te pole is 8. Find er distance from te pole. 6. Heigt of a Tower A water tower is located ft from a building (see te figure). From a window in te building, an observer notes tat te angle of elevation to te top of te tower is 9 and tat te angle of depression to te bottom of te tower is. How tall is te tower? How ig is te window? 9* * ft 6. Determining a Distance An airplane is flying at an elevation of 0 ft, directly above a straigt igway. Two motorists are driving cars on te igway on opposite sides of te plane. Te angle of depression to one car is, and tat to te oter is. How far apart are te cars? 6. Determining a Distance If bot cars in Eercise 6 are on one side of te plane and if te angle of depression to one car is 8 and tat to te oter car is, ow far apart are te cars? 6. Heigt of a Balloon A ot-air balloon is floating above a straigt road. To estimate teir eigt above te ground, te balloonists simultaneously measure te angle of depression to two consecutive mileposts on te road on te same side of te balloon. Te angles of depression are found to be 0 and. How ig is te balloon? 6. Heigt of a Mountain To estimate te eigt of a mountain above a level plain, te angle of elevation to te top of te mountain is measured to be. One tousand feet closer to te mountain along te plain, it is found tat te angle of elevation is. Estimate te eigt of te mountain. 66. Heigt of Cloud Cover To measure te eigt of te cloud cover at an airport, a worker sines a spotligt upward at an angle 7 from te orizontal. An observer 600 m away measures te angle of elevation to te spot of ligt to be. Find te eigt of te cloud cover. 8* * * 7* 600 m Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

9 90 CHAPTER 6 Trigonometric Functions: Rigt Triangle Approac 67. Distance to te Sun Wen te moon is eactly alf full, te eart, moon, and sun form a rigt angle (see te figure). At tat time te angle formed by te sun, eart, and moon is measured to be If te distance from te eart to te moon is 0,000 mi, estimate te distance from te eart to te sun. moon 70. Paralla To find te distance to nearby stars, te metod of paralla is used. Te idea is to find a triangle wit te star at one verte and wit a base as large as possible. To do tis, te star is observed at two different times eactly 6 monts apart, and its apparent cange in position is recorded. From tese two observations E SE can be calculated. (Te times are cosen so tat E SE is as large as possible, wic guarantees tat E OS is 90.) Te angle E SO is called te paralla of te star. Alpa Centauri, te star nearest te eart, as a paralla of Estimate te distance to tis star. (Take te distance from te eart to te sun to be mi.) sun eart E 68. Distance to te Moon To find te distance to te sun as in Eercise 67, we needed to know te distance to te moon. Here is a way to estimate tat distance: Wen te moon is seen at its zenit at a point A on te eart, it is observed to be at te orizon from point B (see te following figure). Points A and B are 6 mi apart, and te radius of te eart is 960 mi. (a) Find te angle u in degrees. (b) Estimate te distance from point A to te moon. eart B A 6 mi moon 69. Radius of te Eart In Eercise 80 of Section 6. a metod was given for finding te radius of te eart. Here is a more modern metod: From a satellite 600 mi above te eart it is observed tat te angle formed by te vertical and te line of sigt to te orizon is Use tis information to find te radius of te eart * O E 0.000* 7. Distance from Venus to te Sun Te elongation a of a planet is te angle formed by te planet, eart, and sun (see te figure). Wen Venus acieves its maimum elongation of 6., te eart, Venus, and te sun form a triangle wit a rigt angle at Venus. Find te distance between Venus and te sun in astronomical units (AU). (By definition te distance between te eart and te sun is AU.) Venus sun AU å eart DISCUSS DISCOVER PROVE WRITE 7. DISCUSS: Similar Triangles If two triangles are similar, wat properties do tey sare? Eplain ow tese properties make it possible to define te trigonometric ratios witout regard to te size of te triangle. S Copyrigt 06 Cengage Learning. All Rigts Reserved. May not be copied, scanned, or duplicated, in wole or in part. Due to electronic rigts, some tird party content may be suppressed from te ebook and/or ecapter(s). Editorial review as deemed tat any suppressed content does not materially affect te overall learning eperience. Cengage Learning reserves te rigt to remove additional content at any time if subsequent rigts restrictions require it.

INTRODUCTION AND MATHEMATICAL CONCEPTS

INTRODUCTION AND MATHEMATICAL CONCEPTS INTODUCTION ND MTHEMTICL CONCEPTS PEVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips of sine,