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08 NCMaths 10 2ed SB TXT.fm Page 276 Thursday, April 14, 2005 3:18 PMStart upWorksheet8-01Brainstarters 81 Express each of these ratios as a fraction.a 5 : 13b 13 : 5e 12 : 13f 13 : 12c 25 : 7g 1:4d 7 : 25h 4:12 Express each of these ratios as a fraction in simplest form.a 20 cm : 1 mb 20 min : 1 hc 15 s : 1 mine 20 cm : 2 mmf 2 mm : 20 cmg 40 s : 1 min3 Calculate the values of the pronumerals in these diagrams.abx d 1 min : 15 sh 1 min : 40 scx 200 130 x 300 defx y 240 x 40 x 315 4 Name the hypotenuse in each of the following right-angled triangles.abBc T8Qwx6CA10yP5 Use Pythagoras’ theorem to find the values of n in each of these right-angled triangles.ab13c63n5.284nn656.56 Solve each of the following:xka --- 7b --- 8.354276NE W CE NT UR Y M AT H S 10: S T AGE S 5.1/ 5.2cd------- 89.7md ---------- 4.912.8

08 NCMaths 10 2ed SB TXT.fm Page 277 Thursday, April 14, 2005 3:18 PMNaming the sides of a right-angled triangleSpecial names are given to the sides of a right-angled triangle. These names depend on theposition of the sides in the triangle relative to a given angle. The hypotenuse is the longest side and is always opposite the right angle. The opposite side is directly opposite the given angle. The adjacent side is next to, or an arm of, the given angle. It runs from the angle to theright angle.For angle θ below, the hypotenuse is OP, theopposite side is XP, the adjacent side is OX.For angle α below, the hypotenuse is OP, theopposite side is OX, the adjacent side is iteOαXoppositeNote: In both triangles, the position of the hypotenuse is fixed while the positions of the oppositeand adjacent sides depend on the position of the angle.In geometry and trigonometry, we use capital letters to label the angles (or vertices) of a triangle,and lower-case letters to label the sides. For example, in ABC we use a to label the side opposite A, b to label the side opposite B, and so on.APcqbBRCarQpExample 1Name the hypotenuse, opposite side and adjacent side for the angle θ in each of these triangles.ab17crθF8qpθGθ15ESolutiona Hypotenuse is 17.Opposite side is 8.Adjacent side is 15.b Hypotenuse is p.Opposite side is r.Adjacent side is q.c Hypotenuse is EF.Opposite side is EG.Adjacent side is FG.TRIGON OMETRY277CHAPTER 8

08 NCMaths 10 2ed SB TXT.fm Page 278 Thursday, April 14, 2005 3:18 PMExample 2For both angles α and β in this triangle, name the hypotenuse, opposite side and adjacent side.247αβ25SolutionFor angle α: hypotenuse is 25 opposite side is 24 adjacent side is 7.For angle β: hypotenuse is 25 opposite side is 7 adjacent side is 24.Just for the recordThe ABC of GreekBelow are six of the letters from the Greek alphabet. (The lower case and capital letters areboth shown.)α, Α alpha β, Β beta γ, Γ gamma θ, Θ theta φ, Φ phi ω, Ω omegaBecause a great deal of our mathematics has come from the ancient Greeks, it is traditional touse Greek letters as pronumerals, particularly in geometry and trigonometry.1 a How many letters are there in the Greek alphabet?b List them and name each one, comparing them to our Roman alphabet.2 Where did the word ‘alphabet’ come from? Explain how it was derived.3 What other alphabets are there?Exercise 8-011 XYZ is rotated about vertex XZYas shown.State whether the followingstatements are true (T) or false (F).a The hypotenuse has remained YXXthe same.b The side opposite angle X is the same.c The side adjacent to X is different.d The side adjacent to Z is the same.e The side opposite Z is different.f In either triangle:i the sides opposite X and Z are differentii the sides adjacent to X and Z are the same.2 Explain why the position of the hypotenuse is fixed in every right-angled triangle.3 Explain how the positions of the opposite and adjacent sides can change in a right-angledtriangle.278NE W CE NT UR Y M AT H S 10: S T AGE S 5.1/ 5.2Z

08 NCMaths 10 2ed SB TXT.fm Page 279 Thursday, April 14, 2005 3:18 PMExample 14 In the following triangles, find:i the hypotenuse (H)ii the side opposite (O) the marked angleiii the side adjacent (A) to the marked angle.ab C13c5wA12dvBeTufx40zyS41R95 In each of the following triangles, find the hypotenuse (H), opposite side (O) and adjacent side(A) for:i angle θii angle φa Dbc12φθφEdφθFSkillBuilder22-01Introduction totrigonometryfe915Example 2θHdeφ36φfcφbIa4527θθθJM6 a For LKM, find the angle:i opposite the hypotenuseii opposite side KMiii opposite side LKiv adjacent to side KMKv adjacent to side LK. Lb Copy and complete these sentences for STU:i Side SU is angle T andto angle S.Uii Side TU is angle S andto angle T.TSTRIGON OMETRY279CHAPTER 8

08 NCMaths 10 2ed SB TXT.fm Page 280 Thursday, April 14, 2005 3:18 PM7 a Sketch a right-angled triangle ABC with hypotenuse AB, and side AC opposite angle θ.b Sketch a right-angled triangle XYZ with hypotenuse YZ, and side XZ adjacent to angle α.c Sketch a right-angled triangle PRQ with side RQ opposite angle RPQ and adjacent to angleQRP. Name the hypotenuse.d Sketch a right-angled triangle DEF, right-angled at E and with the sides opposite andadjacent to angle D equal. What type of triangle is DEF?Worksheet8-02Investigating thetangent rateComparing ratios of sides in similarright-angled trianglesWorking mathematicallyQuestioning and reasoning: Ratios in similar right-angled triangles1 Four similar right-angled triangles are shown below. The shaded angle, A, is 32 (constant) in every triangle.CCCiiAiCBBAiiiivABABa Copy and complete this table, evaluating the ratios (with respect to A) correct to twodecimal places.Ratios (with respect to A)Side length (mm)BCABACBCopposite------ -- ------------------------AChypotenuseBC opposite------ ------------------ABiiiiiiivb What do you notice about the values of the three ratios for the triangles?280NE W CE NT UR Y M AT H S 10: S T AGE S 5.1/ 5.2adjacent

08 NCMaths 10 2ed SB TXT.fm Page 281 Thursday, April 14, 2005 3:18 PM2 a Construct three similar right-angled triangles each with an angle of 60 .FMJ60 HD60 IEb Measure the length of each side, in millimetres.c Copy and complete the table below, evaluatingthe ratios (with respect to 60 ) correct to twodecimal places.K60 LRatios (with respect to 60 te-----------------adjacent DEF HIJ KLMd What can you say about the ratios of sides for any given angle? Give reasons.Using technologyComparing ratios of sides in similar right-angled trianglesCGeometry8-01Ratios of sides inright-angledtrianglesFor more specific instructions click on the technology link.When using dynamic geometry software: to change the size of the smaller triangle, use point D to change the size of the larger triangle, move C or B.EADBTRIGON OMETRY281CHAPTER 8

08 NCMaths 10 2ed SB TXT.fm Page 282 Thursday, April 14, 2005 3:18 PMInstructionsStep 1: Construct the right-angled triangle ACB as shown on the previous page.Step 2: Construct the right-angled triangle DEB inside ACB.AC AC ABDE DE DBStep 3: Measure the ratios -------- , -------- , -------- and compare these to -------- , -------- , -------- .AB BC BCDB BE BEStep 4: Resize each of the triangles and compare the ratios again.1 What do you notice about the ratios?2 Write a statement about the relationship between the corresponding ratios of the sides of thetwo triangles.The trigonometric ratiosoppositeThe ratios of two sides of a right-angled triangle are known as the trigonometric ratios. Thefollowing three trigonometric ratios are the most often used: the sine ratio, which is abbreviated to sin (but still pronounced as sign) the cosine ratio, which is abbreviated to cos the tangent ratio, which is abbreviated to tan.This is how the three trigonometric ratios are defined for agiven angle (θ ) in a right-angled triangle:BoppositeBCaesusin θ --------------------------- -------- or sin θ --no tehypotenuse ABchyp caadjacentACbcos θ --------------------------- -------- or cos θ --hypotenuse ABcθCAopposite BCabtan θ -------------------- -------- or tan θ --adjacentadjacent ACbA useful mnemonic (memory aid) for remembering the three ratios is SOH CAH TOA (pronounced‘so-car-towa’), where: SOH means sine is opposite over hypotenuse CAH means cosine is adjacent over hypotenuse TOA means tangent is opposite over adjacentExample 3In APX, find sin θ, cos θ and tan θ.Aθ135XSolution12PFor angle θ, the hypotenuse is 13, the opposite side is 12 and the adjacent side is 5.oppositeadjacentoppositesin θ --------------------------cos θ --------------------------tan θ 12 cos θ ----- tan θ ----- sin θ -----13135282NE W CE NT UR Y M AT H S 10: S T AGE S 5.1/ 5.2

08 NCMaths 10 2ed SB TXT.fm Page 283 Thursday, April 14, 2005 3:18 PMExample 4a For angle A, find sin A, cos A and tan A.b For angle B, find sin B, cos B, and tan B.CSolutiona For angle A:H is 29O is 20A is 21OAOsin ---- , cos ---- , tan ---HHA20 sin A -----2921cos A -----2920tan A -----21b For angle B:H is 29O is 21A is 202120AB2921 sin B -----2920cos B -----2921tan B -----20Exercise 8-021 Find sin θ, cos θ and tan θ for each of the following triangles:abθ5Example ios264θ5102 Find sin A, cos A and tan A for each of the following .5ABA3 In each of these triangles find the sine, cosine and tangent ratios for:i angle θii angle φab7cθu2524φExample 4FvφφθGθwHTRIGON OMETRY283CHAPTER 8

08 NCMaths 10 2ed SB TXT.fm Page 284 Thursday, April 14, 2005 3:18 PMde84φθ8513fθRθacφbQφS4 Sketch a right-angled triangle, ABC, for each of the following trigonometric ratios. Then findthe length of the third side and write the other two ratios.538a tan A -----b sin B --c cos C -----1251741d sin A --e cos C --f tan B he calculator in trigonometryA scientific calculator can be used to find the trigonometric ratios of angles. We can use thisinformation to find any unknown lengths and angles in a right-angled triangle. The order in whichkeys are used depends on the brand and model of your calculator. Consult your calculator manual orask your teacher for help.Using the calculatorIn trigonometry, angles are usually measured in degrees, minutes and seconds. The keyrelationships are:1 degree 60 minutes1 60′and1 minute 60 seconds1′ 60″An angle of 27 36′15″ is an acute angle of 27 degrees, 36 minutes and 15 seconds. (In size it liesabout halfway between 27 and 28 .) To enter angle sizes involving degrees and minutes into acalculator, use the ′ ″ or DMS (degrees–minutes–seconds) key.Example 5Convert:a 46 45′ to a decimalb 61.87 to degrees, minutes and seconds.Solutiona Enter 46 45′ as follows: ′ ″ 45 ′ ″Press: ′ ″46Display 46 45′ 46.75 b Enter 61.87:61.87 Press: ′ ″ 61.87 61 52′12″284NE W CE NT UR Y M AT H S 10: S T AGE S 5.1/ 5.246 45'00''46.75Display61.8761 52'12''

08 NCMaths 10 2ed SB TXT.fm Page 285 Thursday, April 14, 2005 3:18 PMExample 6Round these angles, correct to the nearest degree.a 48 24′b 64 30′c 15 42′Solution1 60′, so angles with 30 minutes or more are rounded up to the nearest degree.a 48 24′ 48 b 64 30′ 65 c 15 42′ 16 Example 7Round these angles, correct to the nearest minute.a 35 42′18″b 72 35′30″c 39 12′49″Solution1′ 60″, so angles with 30 seconds or more are rounded up to the nearest minute.a 35 42′18″ 35 42′b 70 35′30″ 70′36′c 39 12′49″ 39 13′Example 8Calculate, correct to four decimal places:a cos 76 b tan 57.4 c sin 46 27′SolutionMake sure that your calculator is in the degree mode (DEG) or your answer will be incorrect.Display a760.241 921 cos cos 76 0.2419Display b57.41.563 656 tan tan 57.4 1.5637Display c0.724 773 sin 46 ′ ″ 27 ′ ″ sin 46 27′ 0.7248Example 9Evaluate, correct to two decimal places:a 37 tan 23 Solution a 37b 68.3 sin 38 25′tan 23c23-------------------------cos 18 50′ 37 tan 23 15.705 568 15.71 b 68.3sin 38 ′ ″ 25 68.3 sin 38 25′ 42.439 961 42.44c 23 ′″cos 18 ′ ″ 50 ′ ″ 23 -------------------------- 24.301 038 cos 18 50′ 24.301TRIGON OMETRY285CHAPTER 8

08 NCMaths 10 2ed SB TXT.fm Page 286 Thursday, April 14, 2005 3:18 PMSTAGE5.2Just for the recordDegrees, minutes and secondsThe development of trigonometry began with Hipparchus in Greece around 150 BC, when earlyastronomers tried to solve problems dealing with the positions and apparent movements of thestars and planets. Hipparchus made a table of chords, which today is known as a table of sines.Measuring anglesIn 2000 BC, the Babylonians lived where Iraq is today. The Babylonians invented the units formeasuring angles and time. They believed the Earth took 360 days to travel around the Sunand so a complete revolution was divided into 360 equal parts called degrees. As measuringdevices and calculations required greater precision, each degree was subdivided into 60 equalparts called minutes, and these were further divided into 60 parts called seconds.The word ‘minute’ has different meanings. When pronounced ‘my-newt’, it means tiny, butthis meaning is still related to the minute as a unit of measurement. A minute is a tiny fractionof a degree or hour, and comes from the Latin pars minuta prima, meaning the first (prima)division of a degree or an hour.The word ‘second’ means ‘coming after first’, and this meaning is also related to thesecond as a unit of measurement. Find out how.Exercise 8-03Example 51 Use your calculator to express the following as decimals.a 57 55′b 84 12′c 16 41′e 50 53′f 75 8′g 30 30′25″d 24 30′h 82 40′15″2 Use your calculator to express the following in degrees, minutes and seconds.a 33.76 b 14.1 c 78.15 d 55.5 e 16.7 f 79.23 g 43.9 h 27.07 Example 6Example 73 Round each of the following, correct to the nearest degree.a 56.18 b 56.5 d 27 18′e 27 30′g 33 41′55″h 33 7′5″c 56.75 f 27 54′i 33 14′35″4 Round each of the following, correct to the nearest minute:a 68 39′42″b 68 39′21″d 18 30′27.2″e 18 30′55.3″c 68 39′30″f 18 30′9.5″5 Express each of the following in degrees correct to one decimal place:a 38 14′b 66 7′c 27 11′38″d 45 50′37″e 8 25′11.3″f 81 3′22.5″6 Round each of the following correct to the nearest minute:a 34.45 b 71.087 d 69.4545 e 41.31 Example 8c 5.4829 f 50.213 67 7 Evaluate each of the following, correct to two decimal places:a cos 68 b tan 54 c sin 84 d sin 15 e cos 45 f tan 38 286NE W CE NT UR Y M AT H S 10: S T AGE S 5.1/ 5.2

08 NCMaths 10 2ed SB TXT.fm Page 287 Thursday, April 14, 2005 3:18 PM8 Evaluate each of the following, correct to four decimal places:a sin 23 b cos 60.1 c tan 39.55 d cos 18 24′e tan 75 57′f sin 56 13′9 Calculate each of the following, correct to two decimal places:a 15 sin 46 b 27 tan 37 c 13.5 cos 15 34.5d 14 tan 58e ---------------f 6.7 cos 35sin 80 12084.8g ---------------h 28 cos 17 i ----------------sin 72 tan 68 10 Evaluate, correct to two decimal places:a 12 tan 8 25′b 66.2 cos 81 42′2744.5d ------------------------e ------------------------cos 38.35 tan 65 58′50g 24 tan 36 42′h ---------------sin 70 12.8j ------------------------k 5.3 sin 46.7 cos 46 22′m 19.7 sin 38 45′n 8.9 tan 72.3 Example 9c 18.53 sin 11.813 200f -----------------------sin 54 45′15.7i ---------------sin 30 75.8l ------------------------tan 23 32′o 35.8 cos 87 24′Working mathematicallyApplying strategies and reasoning: Finding an angle, given atrigonometric ratioYou will need: a ruler, a protractor and a calculator.1 a Given that tan A 3--- , measure the size of A to the nearest degree.8Method 1: Use a scale drawing.oppositeSince tan A 3--- -------------------- , construct a right-angled triangle ABC with the8adjacentopposite side to A measuring 3 cm and the adjacent side measuring 8 cm.i Draw a horizontal interval AB of length 8 cm.A8 cmBii At B, draw a perpendicular interval BC of length 3 cm.C3 cmA8 cmBTRIGON OMETRY287CHAPTER 8

08 NCMaths 10 2ed SB TXT.fm Page 288 Thursday, April 14, 2005 3:18 PMiii Join A to C.C3 cmAB8 cmiv Measure the size of A (to the nearest degree).Method 2: Use ‘guess and check’ with a calculator.i Express tan A 3--- as a decimal.8tan A 0.375ii Copy and complete the table to find A.Guesstan AA 40 A 50 A b Repeat the process in part a to find A, given that:i cos A 2--5ii tan A 7-----10iii sin A 0.9c Compare your results with those of other students.2 a Copy the table below and complete it using your calculator (correct to four decimalplaces where necessary).Asin Acos Atan A0 45 90 b The display for tan 90 is ERROR .i Give an explanation for this.ii Find the value of tan 89.9 instead of tan 90 .c Use your table to state whether each of the following is true (T) or false (F).i sin A 0.74ii tan θ 0.95 A is less than 45 θ is less than 45 iii cos Y 0.4183iv sin α 0.93 Y is greater than 45 α is greater than 45 3 Use the ‘guess and check’ method to find B in each case, correct to the nearest degree.a tan B 1.356b sin B 0.2715c tan B 0.3d cos B 2887-----11e sin B NE W CE NT UR Y M AT H S 10: S T AGE S 5.1/ 5.211-----20fcos B 0.0813

08 NCMaths 10 2ed SB TXT.fm Page 289 Thursday, April 14, 2005 3:18 PMUsing the calculator to find anglesA calculator can be used to find an angle if we are given the trigonometric ratio of the angle. Forexample, if sin A 0.76, then angle A can be found by using the inverse sin or sin-1 function.The sin-1 function is activated by keyingshiftsin or INVsin .Worksheet8-03TrigonometriccalculationsExample 10If sin θ 0.65, find the angle θ, correct to the nearest degree.Solutionshift sin 0.65 θ 40.541 θ 41 Example 11STAGEIf tan X 3.754, find the angle X, correct to the nearest minute.Solutionshift tan 3.754 X 75 5′1.62″ X 75 5′5.2 ′″Example 12If cos θ 5--- , find the angle θ, correct to one decimal place.7Solutionshift cos 5 a b / c 7 θ 44.415 θ 44.4 Exercise 8-041 Find the angle θ in each case, correct to the nearest degree.Example 10a cos θ 0.76b tan θ 2.0532c sin θ d tan θ 6e sin θ 7--8fcos θg sin θ h cos θ 1------3itan θ lsin θ 0.4987j1-----10cos θ 0.1352k tan θ 8.8363------2----- 1315SkillBuilders22-15 to 22-16Finding the angle32 For each of the following, find the angle A, correct to the nearest minute.15a tan A -----b sin A 0.815c cos A 7Example 114--55–1---------------4d cos A 0.9387e tan A 4.8fcos A g sin A 5 : 11h sin A 0.88itan A 15.07lsin A jcos A 0.3̇k tan A 2------27--9TRIGON OMETRY289CHAPTER 8

08 NCMaths 10 2ed SB TXT.fm Page 290 Thursday, April 14, 2005 3:18 PMExample 12STAGE5.23 Find the angle X, correct to one decimal place.5a sin X 0.1b cos X ----c sin X 0.71d tan X 4 : 311e sin X 0.4044cos X i2--719-----20ftan X 1.369g cos X h tan X 0.45jtan X 0.502k cos X 3 : 10lsin X 0.6̇Working mathematicallyReasoning and reflecting: The tangent ratio and the gradient of a linerise1 a The gradient, m, of a line is given by: m -------runFind the gradient of each of the lines below, expressing your answers as decimals.iiiyyθθ0iiiyx0θxx0b In each of the diagrams above, θ is the angle the graph line makes with the positivedirection of the x-axis. (This is also called the angle of inclination.) For each line inpart a use a protractor to find the size of θ and then use a calculator to calculate tan θ(correct to three decimal places).c What do you notice about your results for parts a and b?2 For each of the following lines find (as a fraction):riseoppositei m (Remember: m --------)ii tan θ (Remember: tan -------------------- eθadjacentxxriserunWhat do you notice about your results?3 Copy and complete:If θ is the angle between a line and the positive direction of the x-axis, then the g

276 NEW CENTURY MATHS 10: STAGES 5.1/5.2 1 Express each of these ratios as a fraction. a 5:13 b 13:5 c 25:7 d 7:25 e 12:13 f 13:12 g 1:4 h 4:1 2 Express each of these ratios as a fraction in simplest form. a 20 cm:1 m b 20 min:1 h c 15 s:1 min d 1 min:15 s e 20 cm:2 mm f 2 mm:20 cm g 40 s:1 min h 1 min:40 s 3 Calculate the values of the pronumerals in

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