8-4 Sine And Cosine Ratios - Mathematics
8-48-4Sine and Cosine Ratios1. PlanGO for HelpWhat You’ll LearnCheck Skills You’ll Need To use sine and cosine toFor each triangle, find (a) the length of the leg opposite lB and(b) the length of the leg adjacent to lB.determine side lengths intriangles1. 9; 12. . . And Why2.15To use the sine ratio toestimate astronomicaldistances indirectly, as inExample 2B92兹苵苵78B3.73兹苵苵291019197; 2 "7812Lesson 8-3Objectives1Examples1B10; 3 "2923New Vocabulary sine cosine identityTo use sine and cosine todetermine side lengthsin trianglesWriting Sine and CosineRatiosReal-World ConnectionUsing the Inverse of Cosineand SineMath Background1Using Sine and Cosine in TrianglesThe tangent ratio, as you have seen, involvesboth legs of a right triangle. The sine and cosineratios involve one leg and the hypotenuse.Hypotenuseleg opposite /Ahypotenuseleg adjacent to /Acosine of &A hypotenusesine of &A ALeg adjacentto ABLegopposite ACMore Math Background: p. 414DThese equations can be abbreviated:oppositesin A hypotenuseReal-WorldConnectionFor an angle of a given size,the sine and cosine ratios areconstant, no matter wherethe angle is located.1EXAMPLEb. cos Tc. sin Gd. cos GQuick CheckLesson Planning andResourcesadjacentcos A hypotenuseSee p. 414E for a list of theresources that support this lesson.Writing Sine and Cosine RatiosUse the triangle to write each ratio.a. sin TG8sin T hypotenuse 17opposite17adjacentcos T hypotenuse 1517opposite15sin G hypotenuse 17adjacent8cos G hypotenuse 17TR15Present the mnemonic device SOHCAHTOA for thedefinition of the three trigonometric ratios: Sine isOpposite over Hypotenuse; Cosine is Adjacent overHypotenuse; Tangent is Opposite over Adjacent.learning style: verbalCheck Skills You’ll NeedFor intervention, direct students to:Writing Tangent RatiosLesson 8-3: Example 1Extra Skills, Word Problems, ProofPractice, Ch. 8X48ZBelow LevelL1Bell Ringer Practice88064YLesson 8-4 Sine and Cosine RatiosSpecial NeedsPowerPoint641a. sin X 80; cos X 4880 ;48sin Y 80 ; cos Y 64801 a. Write the sine and cosine ratios for&X and &Y. See right.b. Critical Thinking When doessin X cos Y? Explain.sin X cos Y when lX and lYare complementary.A unit circle has radius 1 andcenter (0,0) in the coordinateplane. For all real values of u, thepoint that is reached by travelingu radians from point (1,0) in acounterclockwise direction hascoordinates (cos u, sin u).439L2Have students draw and measure right triangles tomake a table of sine and cosine values for the anglesin the set {10 , 20 , . , 80 }.learning style: visual439
2. TeachOne way to describe the relationship of sine and cosine is to say thatsin x8 cos (90 - x)8 for values of x between 0 and 90. This type of equation iscalled an identity because it is true for all the allowed values of the variable.You will discover other identities in the exercises.Guided Instruction2Visual LearnersPowerPointAdditional Examples1 Use the triangle to find sin T,G20TFor: Sine and Cosine ActivityUse: Interactive Textbook, 8-4sin 22.38 x1x sin 22.38R16sin T 1220 , cos T 20 ,16sin G 20, cos G 1220Quick Check1 AUnot to scaleSolve for x.Use a calculator.2 a. If a 46 for Venus, how far is Venus from the sun in AU?b. About how many miles from the sun is Venus? Mercury?66,960,000 mi; 35,340,000 miabout 0.72 AUWhen you know the leg and hypotenuse lengths of a right triangle, you can useinverse of sine and inverse of cosine to find the measures of the acute angles.3Using the Inverse of Cosine and SineEXAMPLEFind m&L to thenearest degree.16 ftL2.43 A right triangle has a leg 1.5units long and a hypotenuse 4.0units long. Find the measures ofits acute angles to the nearestdegree. 22, 68Problem Solving HintThink of cos-1 (2.44.0 ) as“the angle whose2.4cosine is 4.0,” and3.2sin-1 (4.0) as “theangle whose sine isthe quotient 3.24.0 .”Resources Daily Notetaking Guide 8-4 L3 Daily Notetaking Guide 8-4—L1Adapted InstructionF4403.2OMethod 2cos L 2.44.0Find the trigonometric ratio.sin L 3.24.0m&L cos-1 Q 2.44.0 RUse the inverse.m&L sin-1 Q 3.24.0 RCOS-1 2.4Use a calculator.SIN-1 3.24.0m&L 53Closure4.0Method 153.130102Quick Check4.053.130102m&L 533 Find the value of x. Round your answer to the nearest degree.a.b. 681010x 6.54127x Chapter 8 Right Triangles and TrigonometryAdvanced LearnersEnglish Language Learners ELLL4Encourage students to make conjectures about thevalues of sin 0 , cos 0 , sin 90 , and cos 90 , defendtheir conjectures, and then check the values ona calculator.440SunMercury is about 0.38 AU from the sun.35 A right triangle whose hypotenuseis 18 cm long contains a 65 angle.Find the lengths of its legs to onedecimal place. 16.3 cm, 7.6 cmxa8Use the sine ratio.3794561622.32 A 20-ft wire supporting aflagpole forms a 35 angle withthe flagpole. To the nearest foot,how high is the flagpole?20 ftEarthIf a 22.3 for Mercury, how far is Mercury from thesun in astronomical units (AU)? One astronomicalunit is defined as the average distance from Earth tothe center of the sun, about 93 million miles.1216ConnectionAstronomy The trigonometric ratios have beenknown for centuries by peoples in many cultures.The Polish astronomer Nicolaus Copernicus(1473-1543) developed a method for determiningthe sizes of orbits of planets closer to the sun thanEarth. The key to his method was determiningwhen the planets were in the position shown in thediagram, and then measuring the angle to find a.Have students display a posterlisting the trigonometric ratios.cos T, sin G, and cos G.Real-WorldEXAMPLElearning style: verbalHelp students distinguish between sine and inversesine. The sine of an angle is a ratio, or number. Theinverse sine of a number, or ratio, is an anglemeasure. So, inverses are used to find angle measures.learning style: verbal
EXERCISESFor more exercises, see Extra Skill, Word Problem, and Proof Practice.3. PracticePractice and Problem SolvingAAssignment GuidePractice by ExampleWrite the ratios for sin M and cos M.Example 11.L7K(page 439)GO forHelpExample 2(page 440)25M2472. M94"2 79 ;97 2425 ; 253. KK4兹苵22LL1 A B 1-302兹苵3M41 "32; 2Find the value of x. Round answers to the nearest tenth.4. 11.520x6.56 1017.98.x9.12x1021 106.54.336 (page 440)62 50Find the value of x. Round answers to the nearest degree.2111.514.BApply Your Skills12. 5114x 593.0 x 15.245.813. 46 13x 9x 8516. 66x 4137-4041-47Error Prevention!28 x10. Escalators An escalator in the subway system of St. Petersburg, Russia, has avertical rise of 195 ft 9.5 in., and rises at an angle of 10.48. How long is theescalator? Round your answer to the nearest foot. 1085 ftExample 3Test PrepMixed ReviewTo check students’ understandingof key skills and concepts, go overExercises 2, 14, 17, 22, 27.x117. 17.031-36Homework Quick Check5. 8.3x41 35 C ChallengeExercises 6, 7 Some students mayneed help solving equations withthe variable in the denominator.Review techniques such as crossmultiplication and taking thereciprocal of each side.Exercise 25 Tell students thatthere is also a cotangent ratio.Ask: What do you think is theadjacentcotangent ratio? opposite0.1517x 0.3717. Construction Carlos is planning to builda grain bin with a radius of 15 ft. He readsthat the recommended slant of the roofis 258. He wants the roof to overhangthe edge of the bin by 1 ft. What shouldthe length x be? Give your answer infeet and inches. about 17 ft 8 in.xGPS Guided Problem Solving25 L4Enrichment1 ftoverhangL2ReteachingL1Adapted PracticePracticeNameClass1. 32 and 82. 4 and 163. 11 and 74. 2 and 225. 10 and 206. 6 and 30Algebra Refer to the figure to complete each proportion.sin X20. cos X tanX21. Error Analysis A student states thatBsin A . sin X because the lengths ofYthe sides of #ABC are greater thanthe lengths of the sides of #XYZ.35 Is the student correct? Explain.ZX CNo; the are M and the sine ratio for 35 is constant.ConnectionCorn that fills the bin inExercise 17 would make28,500 gallons of ethanol.x ?7. hy?8. ba h9. ba h?y10. ac ?11. ac h?b ?12. xbopp.adj.hyp.opp. adj. tan XbahxycAlgebra Find the values of the x35 Lesson 8-4 Sine and Cosine Ratios18. sin X 4 cos X hyp. 4Similarity in Right TrianglesAlgebra Find the geometric mean of each pair of numbers.15 ft13.Real-WorldL3DatePractice 8-4Use what you know about trigonometricratios (and other identities) to show thateach equation is an identity. 18–20. See margin.sin X18. tan X cos19. sin X cos X ? tan XXL3A19.20.21.xyxzy331z1yzx2444122. The altitude to the hypotenuse of a right triangle divides thehypotenuse into segments 6 in. and 10 in. long. Find the lengthh of the altitude.opp.20. sin X 4 tan X hyp. 4opp.adj.adj. hyp. cos X19. cos X ? tan X adj. opp.hyp. ? adj.opp. hyp. sin X441
4. Assess & ReteachGOPowerPoint22.GPSB34161. Write the ratios for sin A and30sin B. sin A 1634 , sin B 342. Write the ratios for cos A and30cos B. cos A 34, cos B 1634Use this figure for Exercises3 and 4.xLy57 25a. They are equal; yes;The sine and cosineof complementary' are .25c. Sample: cosine of lA sine of the compl.of lA.27. Yes; use any trig.function and the knownmeasures to find oneother side. Use thePythagorean Thm. tofind the 3rd side.Subtract the acute lmeasure from 90to get the other lmeasure.M3. Find x to the nearest tenth.21.04. Find y to the nearest tenth.13.628e. cos 308 "3 sin 30828f. sin 608 "3 cos 608ProofUse this figure for Exercises5 and 6.N30b – d. Answers mayvary. Samplesare given.30c. sin X 1 forX 89.9; noy 6087x P63R5. Find x to the nearest degree.446. Find y to the nearest degree.46PHSchool.comFor: Graphing calculatorproceduresWeb Code: aue-2111CChallengeAlternative AssessmentHave students write two measurement problems involving distancesin your school. Students also shouldshow how to solve one problemusing the sine ratio and the otherproblem using the cosine ratio.w x 46102x102w42 w xw 37; x 7.5w 68.3; x 151.625. a. In #ABC, how does sin A compare to cos B?BIs this true for the acute angles of other right triangles?3416See left.b. Reading Math The word cosine is derivedfrom the words complement’s sine (see page 694).AC30Which angle in #ABC is the complement of &A?Of &B? lB; lAc. Explain why the derivation of the word cosine makes sense. See left.26. Find each ratio.a. sin P "2b. cos Pa.c. "2b. "2 d. "22222c. sin Rd. cos Re. Make a conjecture about how the sine and cosineof a 45 angle are related. They are equal.P2兹苵127. Writing Leona said that if she had a diagramQRthat showed the measure of one acute angle and thelength of one side of a right triangle, she could find the measure of the otheracute angle and the lengths of the other sides. Is she correct? Explain. See left.28. Find each ratio.1a. sin Sb. cos S 2"31c. sin T a. 2 c.2d. cos T "23e. Make a conjecture about how the sine andcosine of a 30 angle are related. See left.f. Make a conjecture about how the sine andcosine of a 60 angle are related. See left.T2x 30 x兹苵3S60 xN29. For right #ABC with right &C, prove each of the following.a. sin A 1, no matter how large &A is. a–b. See margin.b. cos A 1, no matter how small &A is.30. Graphing Calculator Use the TABLE feature of your graphing calculator to studysin X as X gets close (but 2) to 90. In the Y screen, enter Y1 sin X.a. Use the TBLSET feature so that X starts at 80 and changes by 1. Access theTABLE. From the table, what is sin X for X 89? 0.99985b. Perform a “numerical zoom in.” Use the TBLSET feature, so that X starts with89 and changes by 0.1. What is sin X for X 89.9? 1c. Continue to numerically zoom in on values close to 90. What is the greatestvalue you can get for sin X on your calculator? How close is X to 90? Doesyour result contradict what you are asked to prove in Exercise 29a? See left.d. Writing Use right triangles to explain the behavior of sin X found above.See margin.Show that each equation is an identity by showing that eachBexpression on the left simplifies to 1. 31–34. See margin.c32. (sin B)2 (cos B)2 131. (sin A)2 (cos A)2 1a33.1- (tan A)2 1( cos A) 234.11 1( sin A) 2( tan A) 2CbA35. Show that (tan A)2 - (sin A)2 (tan A)2 (sin A)2 is an identity. See margin.442Chapter 8 Right Triangles and Trigonometry29. Answers may vary.Samples are given.opp.a. Since sin A hyp. ,if sin A L 1, thenopp. L hyp., whichis impossible.44224.10w 3; x 41.4C302523.630 Use this figure for Exercises1 and 2.KHomework HelpVisit: PHSchool.comWeb Code: aue-0804Lesson QuizAFind the values of w and then x. Round lengths to the nearest tenth and anglemeasures to the nearest degree.nlineadj.b. Since cos A hyp. ,if cos A L 1, thenadj. L hyp., whichis impossible.30. d. For ' that approach90, the opp. side getsclose to the hyp. inlength, soopp.hyp.approaches 1.
Real-WorldTest PrepOuter planet’s orbit36. Astronomy Copernicus devised a methoddifferent from the one in Example 2 in orderto find the sizes of the orbits of planets fartherEarth’s orbitfrom the sun than Earth. His method involvednoting the number of days between the timesSunAAthat a planet was in the positions labeled A1 AUand B in the diagram. Using this time and thec8 d8number of days in each planet’s year, heBcalculated c and d.a. For Mars, c 55.2 and d 103.8.How far is Mars from the sun inB not to scaleastronomical units (AU)? about 1.5 AUb. For Jupiter, c 21.9 and d 100.8.How far is Jupiter from the sun in astronomical units? about 5.2 AUConnectionPoland honored Copernicuswith this 1000-zloty note, lastused in 1995.ResourcesFor additional practice with avariety of test item formats: Standardized Test Prep, p. 465 Test-Taking Strategies, p. 460 Test-Taking Strategies withTransparenciesExercise 36 Point out thatCopernicus’s method depends onthe sun, Earth, and outer planets’being in a line at one point intime and forming a right angleat the other point in time.Test PrepMultiple Choice37. What is the value of x to the nearest whole number? AA. 2B. 3C. 4D. 632. (sin B)2 (cos B)2 38. What is the value of y to the nearest tenth? HF. 5.4G. 5.5H. 5.6J. 5.7Short Responseb 2QcR37兹苵苵23 yx39. What is the value of x to the nearest whole number? AA. 53B. 47C. 43D. 3735.1x 40. Use the figure at the right.a. Find m&G. Show your work. a–b. See margin.b. Find m&R by two different methods.Show your work.Gb2 1 a2c233.46.842.3.3743.74xLesson 7-265 30 R45 x44. The wall of a room is in the shape of a golden rectangle. If the height of thewall is 8 ft, what are the possible lengths of the wall to the nearest tenth?12.9 ft or 4.9 ftFind the value of x for each parallelogram.Lesson 6-2845. 3446. 47x - 247. 43x 24x 5lesson quiz, PHSchool.com, Web Code: aua-080431. (sina 2QcR a2 1 b2c2 (cosb 2QcR A)2a22c2 c2 134.1(sin A) 2 2 b2 c145x 6Lesson 8-4 Sine and Cosine Ratios 11(tan A) 21c2a 2 a2Qb R–c 2 2 b2a22 a2 1()() 2– b2 aa35. (tan A)2 – (sin A)2 22a 2– ac 2 a2 – a2 b1818x2bbb2c 2 2 a2 b22bb10Find the value of x. Round answers to the nearest tenth.6.9– (tan A)222 c 2 – a2 1 –41.c 12a 2Qc RLesson 8-3 cMixed ReviewA)21(cos A) 2cc2c2 (1 b2) – a258.5KGO forHelp222 Q ac R b2 a2 443bca2c 2a2b2– 2 2 b2c 2b c22a c 2 a2b2 b2c 2a2 (c 2 2 b2)a2 ? a2 b2c 2a 2 a 2c b(tan A)2(sinb2c 2 ()()A)2740. [2] a. cos G 10mlG 7N 46cos–1 10( )b. mlR N 90 – 46 44 OR mlR 7N 44sin–1 10( )[1] one angle foundcorrectlyc443
Lesson 8-4 Sine and Cosine Ratios 439 Sine and Cosine Ratios The tangent ratio, as you have seen, involves both legs of a right triangle.The sine and cosine ratios involve one leg and the hypotenuse. of &A of &A These equations can be abbreviated: sin A cos A Writing Sine and Cosine Ratios Use the triangle to write each ratio. a. sin T .
Section 5.2 - Graphs of the Sine and Cosine Functions In this section, we will graph the basic sine function and the basic cosine function and then graph other sine and cosine functions using transformations. Much of what we will do in graphing these problems wi
a. Sine, Cosine and Tangent are _ functions that are related to triangles and angles i. We will discuss more about where they come from later! b. We can evaluate a _, _ or _ just like any other expression c. We have buttons on our calculator for sine, cosine and tangent i. Sine ii. Cosine iii. Tangent d.
Explain why the cosine function is a periodic function. 4. Math Journal Draw the graphs for the sine function and the cosine function. Compare and contrast the two graphs. 5. Determine if the function is periodic. If so, state the period. Find each value by referring to the graph of the sine or the cosine function. 6. cos 2 7. sin 5 2 8.
(b) If the power of sine is odd , save one sine factor and use to express the remaining factors in terms of cosine: Then substitute . [Note that if the powers of both sine and cosine are odd, either (a) or (b) can be used.] (c) If the powers of both sine and cosine are even, use the half-ang
cosine curves) of different amplitudes. (Example 1) Since cosine functions are themselves translations of sine functions, any transformation of a cosine function is also a sinusoid by the above definition. There is a special vocabulary used to describe some of our usual gr
Fourier series corresponding to an even function, only cosine terms (and possibly a constant which we shall consider a cosine term) can be present. HALF RANGE FOURIER SINE OR COSINE SERIES A half range Fourier sine or cosine series i
Precalculus: Graphs of Tangent, Cotangent, Secant, and Cosecant The Cotangent Function The tangent function is cotx cosx sinx. It will have zeros where the cosine function has zeros, and vertical asymptotes where the sine function has zeros. It will look like the cosine function where the sine is essentially equal to 1, which is when xis near .
jogensis is the only recorded species of the genus by M.O.P. Iyengar in 1958. It shows the least indulgence of It shows the least indulgence of subsequent researchers in study of this group.
