Let s See What You Already Know

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1 Wha Is This Module Abou? This module will bring you o he world of rigonomery. The word, rigonomery, was aken from he Greek words rigonon and meron. Trigonon means riangle and meron means measure. Thus, our sudy of rigonomery will deal mosly wih riangles. The knowledge and skills ha you will gain in his module will help you solve real-life problems which involve riangles. You will learn many hings in his module wih he following lessons: Lesson 1 Mee Angles Again Lesson 2 Landing on he World of Trigonomery Lesson 3 Degree or Radian? Lesson 4 Your Opposie Is My Adjacen! Wha Will You Learn From This Module? Afer reading his module, you should be able o: classify angles and riangles; find he complemen and supplemen of an angle; and conver he given measuremen of an angle o uni degrees or radians. Wai Before you coninue sudying his module, be sure o have read o module Lines and Angles, a CORE module in he Secondary Level. Le s See Wha You Already Know I. Idenify he ype of angle/riangle being described by he following. 1. An angle is an angle wih a measuremen greaer han 90 bu less han A angle is an angle wih a measuremen of An angle is an angle wih a measuremen greaer han 0 bu less han A riangle wih wo congruen sides is called an riangle. 1

2 5. A riangle wih one righ angle is called a riangle. 6. A riangle wih hree acue angles is called an riangle. 7. A riangle wih equal sides is called an riangle. 8. A riangle wih an obuse angle is called an riangle. II. III. Find he supplemen of each obuse angle and he complemen of each acue angle wih he given measuremens Conver each measuremen o uni radians/degrees. 1. 2π π π Well, how was i? Do you hink you fared well? Compare your answers wih hose in he Answer Key on page 37 o find ou. If all your answers are correc, very good! This shows ha you already know much abou he opic. Who knows, you migh learn a few more new hings as well. If you go a low score, don feel bad. This means ha his module is for you. I will help you o undersand some imporan conceps ha you can apply in your daily life. If you sudy his module carefully, you will learn he answers o all he iems in he es and a lo more! Are you ready? You may go now o he nex page o begin Lesson 1. 2

3 LESSON 1 Mee Angles Again This lesson will help you review he conceps you learned in he module eniled Lines and Angles (CORE module, Secondary Level). Here, you will again mee angles bu you will learn more abou heir differen ypes as well as he so-called complemenary and supplemenary angles. The conceps ha you will learn in his lesson are imporan because we see differen kinds of angles in our environmen. Have you ever seen an arrow poining o a direcion like he one below? This way o he sairs! If you are going o he second floor and you see his arrow, wha will you do? If you answered ha you will follow he direcion he arrow is poining o, you are righ. Le s Learn Do you know ha arrows are also used in mahemaics o represen rays and lines? A ray is formed by a poin of a line and all he poins of he line on one side of he given poin called an endpoin. A ray can go on and on owards he direcion is arrow is poining o. The drawing below is an example of a ray. X Y We shall call i Ray XY. Poin X is is endpoin. 3

4 Rays can mee a a poin. Consider he hands of he clock as in Figure 1 below. C B A Figure 1 Can you sill remember wha you learned in he module eniled Lines and Angles? An angle is formed when wo rays such as AB and AC in Figure 1 mee a a poin such as Poin A in he figure. An angle is formed when wo rays have he same endpoin. The wo rays are called he sides of he angle and he same endpoin is called he verex. BAC has he following sides AB and AC. Le us consider AB as he iniial side and AC as he erminal side. The iniial side of an angle is he side from which you sar measuring. The erminal side, on he oher hand, is where you sop measuring. θ Angles are usually named using hree capial leers of he English alphabe. Bu hey can also be named using oher variables such as numbers or leers of he Greek alphabe. Look a he following examples: BAC, A, 1, Angles are measured using a proracor like he one drawn below. Proracor The unis of measuremens used for angles are degrees ( ) or radians (r). However, he degree is he more commonly used uni. Bu you will also learn how o use radians. Do you remember how angles are classified? We classify angles according o heir measuremen. 4

5 5 Look a he figure below. From Figure 2, he following angles can be seen: BAG, BCH and EFP are acue angles because hey measure less han 90. AGL and GHM are righ angles because hey measure exacly 90. CBA is an obuse angle because i measures more han 90 bu less han 180. IJK is a sraigh angle because i measures 180. L B A G R S M T O H D P C E F N I J K Figure 2 U V G A B F E P B C H A L G G H M C A B K J I

6 RST is a reflex angle because i measures more han 180 bu less han 360 or one complee revoluion. R S T Le s Try This Classify he given angles as, acue, righ, obuse, sraigh or reflex. A E D C B 1. A 2. B 3. C 4. D 5. E Compare your answers wih mine. 1. righ angle 2. obuse angle 3. reflex angle 4. obuse angle 5. obuse angle 6

7 Le s Learn Now, look a he pairs of angles shown in he able below and ake noe of he sums of heir measuremens. Wha are he sums of heir measuremen? If you answered 90, you re righ. The pairs of angles given in he able are complemenary angles. A pair of angles is complemenary if he sum of heir measuremen is 90. In he given able, A and B are complemenary because m A + m B = = 90. We can herefore say ha A is he complemen of B and B is he complemen of A. The oher pairs of angles in he able are also complemenary. Idenify which wo angles among hose lised below complemen each oher. A = 10 B = 80 C = 55 D = 45 1 = 65 2 = 35 Pairs of Angles Measuremen A, B 30, 60 BAC, DEF 45, 45 1, 2 54, 36 α, β 22, 68 If you answered A and B, you are righ. Bu how do we find he measuremen of he complemen of a given angle? Consider he following examples. Example 1 To find he measuremen of he complemen of a given angle, subrac he measuremen of he given angle from 90. Find he measuremen of he complemen of X if m X = 27. Answer: m of he complemen of X = = 63 7

8 Example 2 Wha is he measuremen of he complemen of an angle whose measuremen is 75? Answer: m of he complemen = = 15 Le s Try This Give he measuremens of he complemens of he given angles below. Compare your answers wih hose in he Answer Key on page 37. If you go hem all righ, congraulaions. You did a very good job. Le s Learn Now, le us ge he sums of he measuremens of he pairs of angles Angle in he able below. Measuremen m BOX = Pair of Angles Measuremens m 4 = C, D 30, 150 m Y 30 = 53 RST, WXY 48, 132 m α 48 = , 4 120, 60 m β = 72 ¾ 5, 6 137, 43 Wha do you noice abou he sum of heir measuremens? Their sums are equal o 180 so hey are called supplemenary angles. A pair of angles is supplemenary if he sum of heir measuremens is 180. In he given able, C and D are supplemenary because m C + m D = = 180. We can herefore say ha C is he supplemen of D. The oher pairs of angles in he able are also supplemenary. We can also find he measuremen of he supplemen of a given angle by keeping his in mind: To find he measuremen of he supplemen of a given angle, subrac he measuremen of he given angle from

9 Consider he following examples. Example 1 Wha is he measuremen of he supplemen of 5 if m 5 = 48? Answer: measure of he supplemen of 5 = = 132 Example 2 Find he measuremen of he supplemen of an angle whose measure is 123. Answer: measuremen of he supplemen = = 57 Le s Try This Fill up he able below. Angle Measurem m RST = m 5 = m X = Compare your answers wih hose in he Answer Key on page 38. Coninue reading his module if you go hem all righ. If no, review he pars you m did α no undersand = well. Le s See Wha You Have Learned m β = 176 ¾ I. Classify he given angles as o acue, righ, obuse, sraigh or reflex. Wrie your answers in he blanks. 1. s A E s K 9

10 s D A E s 1 s G F s H I s G s H W s E s 4. X 7. Y s F G s II. III. Solve for he measuremens of he complemens of he following angles given heir measuremens Solve for he measuremens of he supplemens of he following angles given heir measuremens Compare your answers wih hose in he Answer Key on page 38. If you go a score of 13 or higher, very good. Coninue reading his module. If you go below 13, read he lesson again and solve more problems similar o he ones in his module. 10

11 Le s Remember An angle is formed by wo rays wih a common endpoin. The wo rays are called he sides of he angle and he common endpoin is called he verex. Angles are usually named using hree capial leers of he English alphabe. Bu hey can also be named using numbers or leers of he Greek alphabe. A proracor is an insrumen used o measure an angle. The unis of measuremen for angles are degrees ( ) or radians (r). Angles are classified according o heir measuremens. Acue angle an angle whose measuremen is less han 90. Righ angle an angle whose measuremen is exacly 90. Obuse angle an angle whose measuremen is more han 90 bu less han 180. Sraigh angle an angle whose measuremen is exacly 180. Reflex angle an angle whose measuremen is more han 180 bu less han 360. If he sum of he measuremens of wo angles is 90, hey are complemenary angles. To find he measuremen of he complemen of a given angle, subrac he measuremen of he given angle from 90. If he sum of he measuremens of wo angles is 180 hey are supplemenary angles. To find he measuremen of he supplemen of a given angle, subrac he measuremen of he given angle from

12 LESSON 2 Landing on he World of Trigonomery Afer learning all abou angles in he previous lesson, you are now ready o ackle more complicaed figures such as riangles. This lesson will focus on he differen kinds of riangles. The knowledge you will gain from his lesson will hopefully help you in solving real-life problems. Le s Sudy and Analyze A B C D E F Look a he differen figures above. Which of hem are riangles? Figures B,C,E and F are riangles. Wha is a riangle? A riangle is a polygon wih hree sides and hree angles. Consider he riangle below, ABC. A B C 12

13 The sides of ABC are AB, AC and BC. Name he angles ha make i up. If your answers are A, B and C, you are righ. Bu riangles are of differen kinds. Triangles can be classified according o he lenghs of heir sides. Figure 1 Figure 2 Figure 3 Figure 1 shows a riangle wih no congruen sides. I is called scalene riangle. Figure 2 shows a riangle wih wo congruen sides. I is called an isosceles riangle. Two sides are congruen if hey are equal. Figure 3 shows a riangle wih hree congruen sides. I is called an equilaeral riangle. A scalene riangle is a riangle wih no congruen sides. An isosceles riangle is a riangle wih wo congruen sides. An equilaeral riangle is a riangle wih hree congruen sides. Le s Try This Idenify wheher each riangle below is a scalene, an isosceles or an equilaeral riangle

14 Compare your answers wih mine. 1. isosceles 2. isosceles 3. scalene 4. scalene 5. equilaeral 6. equilaeral 7. equilaeral Did you ge all he answers righ? If you did, ha s good. Proceed o he nex par of he lesson. Le s Learn Look a anoher se of riangles below. This shows anoher way of classifying riangles. From his, you will see ha riangles can be classified also according o he kinds of angles ha make hem up. Figure 4 Figure 5 Figure 6 Figure 4 shows a riangle wih a righ angle. I is called a righ riangle. Figure 5 shows a riangle made up of hree acue angles. I is called an acue riangle. Figure 6 shows a riangle wih an obuse angle. I is called an obuse riangle. Le s Try This Deermine wheher he following riangles are acue, righ or obuse. Wrie your answers in he blanks

15 4. 5. Compare your answers wih hose in he Answer Key on page 38. Le s See Wha You Have Learned A. Fill in he blanks wih he correc answers. 1. A riangle wih an obuse angle is called an riangle. 2. An isosceles riangle has congruen sides. 3. A riangle wih hree acue angles is called an riangle. 4. A riangle has no congruen sides. 5. If all he sides of a riangle are congruen, i is herefore an riangle. 6. A riangle wih wo congruen sides is called an riangle. 7. A riangle wih a righ angle is called a riangle. 8. If he wo sides of a riangle are perpendicular, he riangle is herefore a riangle. B. Based on he given figure, classify he riangles lised afer i. C D A B J K I H E F M L 1. ABC 2. DEF G 3. JIF 15

16 4. FGH 5. FMG 6. IKL 7. FHM Compare you answers wih hose in he Answer Key on page 39. If you go a score of 13 o 15, you re doing grea. Coninue reading his module. If you go a score of below 13, read his lesson again and solve more problems similar o hose given before proceeding o he nex lesson. Le s Remember A riangle is a polygon wih hree sides and hree angles. Two sides of a riangle are congruen if hey have he same measuremen or hey are equal. Triangles can be classified according o: A. he lenghs of heir sides and 1. Scalene riangle riangle wih no congruen sides 2. Isosceles riangle riangle wih wo congruen sides 3. Equilaeral riangle riangle wih hree congruen sides B. kinds of angles ha he riangle has. 1. Acue riangle riangle wih hree acue angles 2. Righ riangle riangle wih a righ angle 3. Obuse riangle riangle wih an obuse angle 16

17 LESSON 3 Degree or Radian? In his lesson, you will learn he difference beween he wo unis of measuremen used for angles: degrees and radians. You will also learn how o conver one uni o anoher. This lesson is imporan because here are problems which you migh mee in he fuure which involve measuremen of angles. Le s Sudy and Analyze Look a he drawing of a park above. Wha do you noice abou is shape? I is circular in shape. Suppose you walk around he park? Wha do you call he pah you will ake? This pah is equivalen o he disance around he park. I is called he circumference of he park. The circumference is he disance around a circle. I is always equal o 360 uni (eiher or radian (rad)). Do you know wha an arc is? I is porion of he circumference of a circle. Suppose you ake a walk around ha park wih your grandfaher, moher, faher and siser. Your grandfaher was no able o cover he whole circumference of he park. He was only able o cover ¼ of he circumference. How far was he able o walk? ¼ of he park s circumference is ¼ of 360 unis = 90 unis Your moher was able o cover ½ of he park s circumference. How far was she able o work? ½ of he park s circumference is ½ of 360 unis = 180 unis 17

18 Your faher was able o cover ¾ of he park s circumference. How far was he able o walk? ¾ of he park s circumference is ¾ of 360 unis = 270 unis Your siser covered he shores disance, from a poin on he circumference of he park o he cener of he park. She only covered he radius of he park. How far was she able o walk? This will be he focus of he following discussion. Le s Learn Look a he circle drawn below. circumference A degree is a uni of measuremen equal o 1/360 of he circumference of a circle. From his, we can conclude ha: 90 = 90 (1/360) of he circumference = ¼ of he circumference 180 = 180 (1/360) of he circumference = ½ of he circumference 270 = 270 (1/360) of he circumference = ¾ of he circumference An arc is par of a circle. arc 18

19 A radius is a line drawn from he cener of he circle o any poin on he circle. radius arc A radian is an angle which, if placed a he cener of he circle, makes an arc equal o he radius of he circle. radius arc In he figure drawn above, he lengh of he radius of he circle is equal o he lengh of he arc formed by he wo radii (plural of radius). This means ha he measure of he angle formed, m 1 = 1 radian. The circumference of a circle = 2 π radians. π is a symbol which has a consan value. ( π 180 ) π is read as approximaely equal o. I means ha he value is rounded off. Below are some examples of measures in radians. π 2 rad π rad 0 or 2π rad There are rad or rad in 1. So, o conver degrees o radians, we muliply he number of degrees by ( π 180 ) or Look a he examples below. Example 1 30 = 30 ( π 180 ) = π 6 rad 3π 2 rad 19

20 Example 2 60 = 60 ( π 180 ) = π 3 rad Example 3 79 = 79 ( ) = rad Le s Try This Conver he following degree measuremens ino radians = 120 ( π 180 ) = 2π = 135 = 3π = 225 ( π 180 ) = = = = = = = Compare your answers wih hose in he Answer Key on page 39. If you go all he answers righ, congraulaions! If you made some misakes, don worry, jus read he discussion again. Le s Learn There are 180 π or in 1 rad. Thus, o conver radians o degrees, we muliply he number of radians by 180 π or Sudy he following examples. Example 1 π 4 rad = 4 Example 2 π ( π ) 5π 6 rad = 6 Example = 180/4 = 45 5π ( π ) 180 = rad =.36 ( ) =

21 Le s Try This Conver he following radian measuremens ino degrees. 1. π 3 rad = π 4 rad = 4 π ( π ) 180 = 60 3π ( π ) 180 = 3. 7π 6 rad = = 4. 5π 3 rad = = 5..5 rad = = Compare your answers wih mine. 2. 3π 4 rad = π 6 rad = π 3 rad = 3 3π ( π ) 180 = 135 7π ( π ) 180 = 210 5π ( π ) 180 = rad =.5 ( ) = If you go all he answers righ, ha means you undersand he lesson well. If you did no, go back o he pars of he discussion you did no undersand very well. Le s See Wha You Have Learned Mach he iems in column A wih heir equivalens in column B. A B a b. π c d e. π f g π 2 h π 3 i. 4π π 4 j. 5π 4 21

22 11. 5π 3 k. 7π π 6 l π 4 m. π 4 n. π 3 Compare your answers wih hose in he Answer Key on pages Le s Remember A degree is equal o 1/360 of he circumference of a circle. A radian, if placed a he cener of a circle, makes an arc equal o he radius of he circle. π 180 rad or rad = 1 To conver degrees o radians, we muliply he number of degrees by π 180 or π or = 1 rad. To conver radians o degrees, we muliply he number of radians by 180 π or

23 LESSON 4 Your Opposie Is My Adjacen! Do you know wha he sin, cos and an keys in your calculaor are for? This is wha his lesson will be abou. Many real-life problems involve righ riangles. This lesson will herefore each you wha rigonomeric funcions are and how hey are relaed o righ riangles. Le s Sudy and Analyze 5 unis 4 unis A =70 3 unis Wha kind of riangle was formed in he figure above? A righ riangle was formed. Le us look a A which measures 70. Wha is drawn opposie A? A ree is drawn opposie i. How was A formed? I was formed by a line segmen drawn from he foo of he ree o he sone and anoher line segmen from he sone o he op of he ree. The line segmen drawn from he foo of he ree o he sone is said o be adjacen o A. The line segmen drawn from he sone o he op of he ree is called he hypoenuse of he righ riangle. Wha is he heigh of he ree in he figure? Wha abou he disance of he sone from he foo of he ree? Wha is he lengh of line segmen drawn from he sone o he op of he ree? The heigh of he ree is 4 unis. The disance of he sone from he foo of he ree is 3 unis. The lengh of he line segmen drawn from he sone o he op of he ree is 5 unis. 23

24 Are all your answers righ? If hey are, very good! Compue he following raios. heigh of he ree 4 unis = hypoenuse 5 unis disance from he ree o hesone hypoenuse heigh of he ree = disance from he ree o he sone = Are your answers If hey are, you are righ. Le s Try This 6.4 unis h = 5 unis 3 unis 4 unis and? 5 unis 3 unis X 55 shadow 4 unis Le us look a he angle which measures 55 in he figure above. We will call his angle X. Wha is opposie X? The heigh of he man is opposie i. Wha formed X? X is formed by he shadow of he man and he line segmen from he ip of he shadow o he head of he man. The man s shadow is adjacen o X. Le us lis down he given measuremens: heigh of he man lengh of he man s shadow hypoenuse = 5 unis = 4 unis = 6.4 unis 24

25 Now, le us lis down he hree raios: heigh of he man = hypoenuse lengh of he man's shadow = hypoenuse heigh of he man lengh of his shadow = 5 4 Le s Learn Look a he righ riangle drawn below. B 5 unis 13 unis A C 12 unis A Figure 1 Le us measure he lengh of each side of he righ riangle. AB = 13 unis BC = 5 unis AC = 12 unis There is a special name for each side of a righ riangle. The wo sides of ABC ha form he righ angle, AC and BC, are called he legs of he riangle. The hird side, AB, is called is hypoenuse. The hypoenuse is he side opposie he righ angle. The riangle in Figure 1 has wo acue angles, and B. Each of hem is formed by he hypoenuse and one of he legs. For example, A is formed by he hypoenuse AB and he leg AC. The leg ha helps form an acue angle is said o be adjacen o ha angle. In my example, AC is he leg adjacen o A. 25

26 The same leg is said o be opposie he oher acue angle, B. We say AC is opposie B. Again, in Figure 1: AB is he hypoenuse BC is opposie A AC is adjacen o A Le s Try This Look a he following figures. Deermine he hypoenuse, he side opposie he given angle and he side adjacen o he given angle of each riangle. X W Figure 2 12 Y hypoenuse = XY side opposie Y = WX side adjacen o Y = WY side opposie X = WY side adjacen o X = WX D E 24 Figure 3 F hypoenuse = DF side opposie D = EF 26

27 side adjacen o D = DE side opposie F = side adjacen o F = T 15 9 R 12 Figure 4 S hypoenuse = side opposie R = side adjacen o R = side opposie T = lengh of side opposie A sin A = lengh of hypoenuse side adjacen o T = Compare your answers wih hose in he Answer Key on page 40. Le s Learn Afer discussing he differen pars of a righ riangle, you are now ready o learn abou he various rigonomeric funcions. These are used in solving for unknown measuremens of pars of righ riangles. Sine of an Angle The sine of A is he raio of he lengh of he side opposie A o he lengh of he hypoenuse. The sine of A is abbreviaed as sin A. Thus, we have: sin A = opposie hypoenuse 27

28 Look a Figures 5, 6, 7 and 8 below. B X C 12 Figure 5 A W Figure 6 12 Y D E 24 Figure 7 F T 15 9 R 12 Figure 8 S In Figure 5, sin A = BC/AB = 5/13; sin B = AC/AB = 12/13 In Figure 6, sin X = WY/XY = 12/25; sin Y = WX/XY = 15/25 In Figure 7, sin F = = ; sin D = = In Figure 8, sin T = = ; sin R = = Compare your answers wih mine. Figure 7, sin F = DE/DF = 10/26; sin D = EF/DF = 24/26 Figure 8, sin T = RS/RT = 12/15; sin R =TS/RT = 9/15 28

29 Cosine of an Angle The cosine of A is he raio of he lengh of he side adjacen o A and he lengh of he hypoenuse. The cosine of A is abbreviaed as cos A. Thus, we have: lengh of side adjacen o A cos A = hypoenuse cos A = adjacen hypoenuse In Figure 5, cos A = AC/AB = 12/13; cos B = BC/AB = 5/13 In Figure 6, cos X = WX/XY = 15/25; cos Y = WY/XY = 12/25 In Figure 7, cos F = = ; cos D = = In Figure 4, cos T = = ; cos R = = Compare your answers wih mine. Figure 7, cos F = EF/DF = 24/26; cos D = DE/DF = 10/26 Figure 8, cos T = ST/RT = 9/15; cos R = RS/RT = 12/15 Tangen of an Angle The angen of A is he raio of he lengh of he side opposie A and he lengh of he side adjacen o A. Tangen A is abbreviaed as an A. Thus, we have: lengh of side opposie A an A = lengh of side adjacen o A opposie an A = adjacen In Figure 5, an A = BC/AC = 5/12; an B = AC/BC = 12/5 In Figure 6, an X = WY/WX = 12/15; an Y = WX/WY = 15/12 In Figure 7, an F = = ; an D = = In Figure 8, an T = = ; an R = = Compare your answers wih mine. Figure 7, an F = DE/EF = 10/24; an D = EF/DE = 24/10 Figure 8, an T = RS/ST =12/9; an R = ST/RS = 9/12 29

30 Coangen of an Angle The coangen of A is he reciprocal of he angen of A. I is he raio of he lengh of he side adjacen o A and he lengh of he side opposie A. The coangen of A is abbreviaed as co A. Thus, we have: lengh of side adjacen o A co A = lengh of side opposie A adjacen co A = opposie In Figure 5, co A = AC/BC = 12/5; co B = BC/AC = 5/12 In Figure 6, co X = WX/WY = 15/12; co Y = WY/WX = 12/15 In Figure 7, co F = = ; co D = = In Figure 8, co T = = ; co R = = Compare your answers wih mine. Figure 7, co F = EF/DE = 24/10; co D = DE/EF = 10/24 Figure 8, co T = ST/RS = 9/12; co R = RS/ST = 12/9 Secan of an Angle The secan of A is he reciprocal of he cosine of A. I is he raio of he lengh of he hypoenuse and he lengh of he side adjacen o A. The secan of A is abbreviaed as sec A. Thus, we have: lengh of hypoenuse sec A = lengh of side opposie A hypoenuse sec A = adjacen In Figure 5, sec A = AB/AC = 13/12; sec B = AB/BC = 13/5 In Figure 6, sec X = XY/WX = 25/15; sec Y = XY/WY = 25/12 In Figure 7, sec F = = ; sec D = = In Figure 8, sec T = = ; sec R = = Compare your answers wih mine. In Figure 7, sec F = DF/EF = 26/24; sec D = DF/DE = 26/10 In Figure 8, sec T = RT/ST = 15/9; sec R = RT/RS = 15/12 30

31 Cosecan of an Angle The cosecan of A is he reciprocal of he sine of A. I is he raio of he lengh of he hypoenuse and he lengh of he side opposie A. The cosecan of A is abbreviaed as csc A. Thus, we have: lengh of hypoenuse csc A = lengh of side opposie A hypoenuse csc A = opposie In Figure 5, csc A = AB/BC = 13/5; csc B = AB/AC = 13/12 In Figure 6, csc X = XY/WY = 25/12; csc Y = XY/WX = 25/15 In Figure 7, csc F = = ; csc D = = In Figure 8, csc T = = ; csc R = = Compare your answers wih mine In Figure 7, csc F = DF/DE = 26/10; csc D = DF/EF = 26/24 In Figure 8, csc T = RT/RS = 15/12; csc R = RT/RS = 15/9 Did you ge all he answers righ? If you did, ha s very good. If you did no, ry going over he lesson again. Le s See Wha You Have Learned For each given figure, deermine he raios lised. 1. A C B a. sin A b. sec B c. an B d. co A 31

32 2. W X Y a. cos W b. csc X c. an X d. co W 3. D E F a. co D b. an E c. sec D d. sin E 4. M F C a. sin M b. sec M c. co F d. cos F 32

33 Compare your answers wih hose in he Answer Key on pages Le s Remember The wo sides of he riangle ha form he righ angle are called is legs. The hird side is called is hypoenuse. The leg ha helps form an acue angle in a righ riangle is said o be adjacen o ha angle. The same leg is said o be opposie he oher acue angle. The hypoenuse is always he side opposie he righ angle. The six rigonomeric funcions are defined as follows: lengh of side opposie A sin A = lengh of hypoenuse = opposie hypoenuse lengh of side adjacen o A cos A = lengh of hypoenuse lengh of side opposie A an A = lengh of side adjacen o A = adjacen hypoenuse opposie = adjacen co A = sec A = lengh of side adjacen o A lengh of side opposie A lengh of hypoenuse lengh of sideadjacen A adjacen = opposie hypoenuse = adjacen lengh of hypoenuse csc A = lengh of side opposie A hypoenuse = opposie Now ha you have finished reading he module, are you ready o know how much you have learned? Bu before doing ha, read he module summary firs o review wha you have learned. Le s Sum Up In his module, you learned ha: An angle is formed by wo rays wih a common endpoin. The wo rays are called he sides of he angle and he common endpoin is called is verex. Angles are usually named using, hree capial leers of he English alphabe. Bu oher variables such as a number or a Greek leer can also be used. 33

34 Angles are measured using a proracor. The unis of measuremen used for angles are degree ( ) or radian (rad). Angles can be classified according o heir measuremens: Acue angle an angle which measures less han 90 Righ angle an angle which measures exacly 90 Obuse angle an angle which measures more han 90 bu less han 180 Sraigh angle an angle which measures exacly 180 Reflex angle an angle which measures more han 180 bu less han 360 If he sum of he measuremens of wo angles is 90, hey are complemenary. To find he measuremen of he complemen of a given angle, subrac is measuremen from 90. Two angles whose sum of measuremens is 180 are called supplemenary angles. To find he measuremen of he supplemen of a given angle, subrac is measuremen from 180. A riangle is a polygon wih hree sides and hree angles. Congruen means having he same measuremen. Congruen sides/angles have he same measuremens. Triangles can be classified according o: A. Congruence of sides 1. Scalene riangle a riangle wih no congruen sides 2. Isosceles riangle a riangle wih wo congruen sides 3. Equilaeral riangle a riangle wih hree congruen sides B. Kinds of angles ha he riangle has 1. Righ riangle a riangle wih one righ angle 2. Acue riangle a riangle wih hree acue angles 3. Obuse riangle a riangle wih one obuse angle A degree is equal o 1/360 of he circumference of a circle. A radian, if placed a he cener of he circle, makes an arc equal o he radius of he circle. π 180 rad or rad = 1 To conver degree measuremen ino radians, we muliply he number of degrees by π 180 or

35 180 / π or = 1 rad. To conver radian measuremens ino degrees, we muliply he number of radians by 180 /π or The wo sides of he riangle ha form he righ angle are called is legs. The hird side is called is hypoenuse. The leg ha helps form an acue angle in a righ riangle is said o be adjacen o ha angle. The same leg is said o be opposie he oher acue angle. The hypoenuse is always he side opposie he righ angle. The six rigonomeric funcions are: lengh of side opposie A sin A = lengh of hypoenuse = opposie hypoenuse lengh of side adjacen o A cos A = lengh of hypoenuse lengh of side opposie A an A = lengh of side adjacen o A = adjacen hypoenuse opposie = adjacen co A = sec A = lengh of side adjacen o A lengh of side opposie A lengh of hypoenuse lengh of sideadjacen A adjacen = opposie hypoenuse = adjacen lengh of hypoenuse csc A = lengh of side opposie A hypoenuse = opposie Wha Have You Learned? I. Wrie rue if he saemen is correc and false if i is no. 1. A riangle wih one obuse angle is an acue riangle. 2. A reflex angle measures more han 90 bu less han A righ riangle has one righ angle. 4. A riangle is equilaeral if all is angles are congruen An isosceles riangle has no congruen sides. An acue angle is bigger han an obuse angle. 35

36 II. III. 7. A reflex angle is congruen o an obuse angle. 8. All he angles of an acue riangle are less han 90. Solve for he complemen of each angle and express your answer in rad Conver each given measuremen ino degrees hen find he given angle s supplemen. 1. 5π /6 rad 2. π /4 rad 3. 2π /3 rad 4. π /6 rad 5. 3π /4 rad Compare your answers wih hose in he Answer Key on page

37 Answer Key A. Le s See Wha You Already Know (pages 1 2) I. 1. obuse 2. sraigh 3. acue 4. isosceles 5. righ 6. acue 7. equilaeral 8. obuse II = = = = = 15 III π ( π ) π 180 = 120 7π ( π ) 180 = ( π ) 180 = ( π 180 ) = π ( π 180 ) = 2π ( π 180 ) = 11π 6 B. Lesson 1 Le s Try This (page 8) Angle Measuremen of m BOX = = 58 m 4 = = 43 m Y = = 37 m α = = 26 m β = 72 ¾ ¾ = 17 37

38 Le s Try This (page 9) Le s See Wha You Have Learned (pages 9 10) I. 1. sraigh 2. righ 3. acue 4. obuse 5. reflex 6. obuse 7. righ Angle II = = = = 16 Measuremen m RST = = m 5 = = m X = = m α = = m β = 176 ¾ ¾ = III = = = = 38 C. Lesson 2 Le s Try This (pages 14 15) 1. righ 2. obuse 3. obuse 4. acue 5. acue 38

39 Le s See Wha You Have Learned (pages 15 16) A. 1. obuse 2. wo 3. acue 4. scalene 5. equilaeral 6. isosceles 7. righ 8. righ B. 1. isosceles because AC and BC are congruen 2. righ because E is a righ angle 3. equilaeral because he hree sides are congruen 4. acue because all of is angles are acue 5. righ 6. equilaeral 7. righ D. Lesson 3 ( π 180 ) Le s Try This (page 20) = 120 = 2π = 135 ( π 180 ) = 3π = 225 ( π 180 ) = 5π = 240 ( ) = rad = 315 ( ) = rad = 330 ( ) = 5.76 rad Le s See Wha You Have Learned (pages 21 22) 1. (b) 30 ( π 180 ) = π /6 2. (m) 45 ( π 180 ) = π /4 39

40 3. (n) 60 ( π 180 ) = π /3 4. (e) 180 ( π 180 ) = π 5. (k) 210 ( π 180 ) = 7π /6 6. (j) 225 ( π 180 ) = 5π /4 7. (i) 240 ( π 180 ) = 4π /3 8. (a) 2 9. (l) (g) (h) (c) (f) 4 π ( π ) 180 = 90 2π ( π ) 180 = 120 3π ( π ) 180 = 135 5π ( π ) 180 = π ( π ) 180 = 330 7π ( π ) 180 = 315 E. Lesson 4 Le s Try This (pages 26 27) Figure 3 side opposie F = DE side adjacen o F = EF Figure 4 hypoenuse = RT side opposie R = ST side adjacen o R = RS side opposie T = RS side adjacen o T = ST Le s See Wha You Have Learned (pages 31 32) 1. a. sin A = BC/AB b. sec B = AB/BC c. an B = AC/BC d. co A = AC/BC 2. a. cos W = WY/WX b. csc X = WX/WY 40

41 c. an X = WY/XY d. co W = WY/XY 3. a. co D = DF/EF b. an E = DF/EF c. sec D = DE/DF d. sin E = DF/DE 4. a. sin M = CF/FM b. sec M = FM/CM c. co F = CF/CM d. cos F = CF/FM F. Wha Have You Learned? (pages 35 36) I. 1. false, i is an obuse riangle 2. false, i measures more han 180 bu less han rue 4. rue 5. false, i has wo congruen sides 6. false, i is smaller 7. false, a reflex is bigger han an obuse angle 8. rue II = 45 ; 45 = π = 0 ; = 0 rad ( π 180 ) π 180 = π = 30 ; 30 ( ) = 15 ; 15 ( ) = = 60 ; 60 ( π 180 ) = π 3 III. 1. 5π /6 rad ( π ) 2. π /4 rad ( π ) 3. 2π /3 rad ( π ) 4. π /6 ( π ) 5. 3π /4 ( π ) 180 = 150 ; = = 45 ; = = 120 ; = = 30 ; = = 135 ; = 45 References Sia, Lucy O., e al. 21 s Cenury Mahemaics, Second Year. Quezon Ciy: Phoenix Publishing House, Inc. Reprined Capiulo, F.M. Algebra, a Simplified Approach. Manila: Naional Booksore,

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