# Section 5-5 Solving Right Triangles*

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 5-5 Solving Right Tringles Geometry. The re of retngulr n-sided polygon irumsried out irle of rdius is given y A n tn 80 n (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute eh to five deiml ples. (B) Wht numer does A seem to pproh s n? (Wht is the re of irle with rdius?) 80. Geometry. The re of regulr n-sided polygon insried in irle of rdius is given y A n 360 sin 2 n n 8 r (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute eh to five deiml ples. (B) Wht numer does A seem to pproh s n? (Wht is the re of irle with rdius?) 8. Angle of Inlintion. Rell (Setion 2-) the slope of nonvertil line pssing through points P (x, y ) nd P 2 (x 2, y 2 ) is given y slope m (y 2 y )/(x 2 x ). The ngle tht the line L mkes with the x xis, 0 80, is lled the ngle of inlintion of the line L (see figure). Thus, Slope m tn, 0 80 (A) Compute the slopes to two deiml ples of the lines with ngles of inlintion 88.7 nd (B) Find the eqution of line pssing through (4, 5) with n ngle of inlintion 37. Write the nswer in the form y mx, with m nd to two deiml ples. L y 82. Angle of Inlintion Refer to Prolem 8. (A) Compute the slopes to two deiml ples of the lines with ngles of inlintion 5.34 nd (B) Find the eqution of line pssing through (6, 4) with n ngle of inlintion 06. Write the nswer in the form y mx, with m nd to two deiml ples. L x Setion 5-5 Solving Right Tringles* FIGURE In the previous setions we hve pplied trigonometri nd irulr funtions in the solutions of vriety of signifint prolems. In this setion we re interested in the prtiulr lss of prolems involving right tringles. A right tringle is tringle with one 90 ngle. Referring to Figure, our ojetive is to find ll unknown prts of right tringle, given the mesure of two sides or the mesure of one ute ngle nd side. This is lled solving right tringle. Trigonometri funtions ply entrl role in this proess. To strt, we lote right tringle in the first qudrnt of retngulr oordinte system nd oserve, from the definition of the trigonometri funtions, six trigonometri rtios involving the sides of the tringle. [Note tht the right tringle is the referene tringle for the ngle.] *This setion provides signifint pplition of trigonometri funtions to rel-world prolems. However, it my e postponed or omitted without loss of ontinuity, if desired. Some my wnt to over the setion just efore Setions 7- nd 7-2.

2 380 5 TRIGONOMETRIC FUNCTIONS TRIGONOMETRIC RATIOS (, ) 0 90 sin os tn s se ot Side is often referred to s the side opposite ngle, s the side djent to ngle, nd s the hypotenuse. Using these designtions for n ritrry right tringle removed from oordinte system, we hve the following: RIGHT TRIANGLE RATIOS Hyp Opp sin Opp Hyp os Adj Hyp s Hyp Opp se Hyp Adj Adj 0 90 tn Opp Adj ot Adj Opp Explore/Disuss For given vlue, 0 90, explin why the vlue of eh of the six trigonometri funtions is independent of the size of the right tringle tht ontins. TABLE Angle to Nerest Signifint Digits for Side Mesure 2 0 or 0. 3 or or The use of the trigonometri rtios for right tringles is mde ler in the following exmples. Regrding omputtionl ury, we use Tle s guide. (The tle is lso printed inside the k over of this ook for esy referene.) We will use rther thn in mny ples, relizing the ury indited in Tle is ll tht is ssumed. Another word of ution: when using your lultor e sure it is set in degree mode.

3 5-5 Solving Right Tringles 38 EXAMPLE Right Tringle Solution Solve the right tringle with 6.25 feet nd Solution First drw figure nd lel the prts (Fig. 2): FIGURE ft 32.2 Solve for nd re omplementry. Solve for sin Or use s. sin sin feet Solve for os Or use se. os os feet MATCHED PROBLEM Solve the right tringle with 27.3 meters nd In the next exmple we re onfronted with prolem of the type: Find given sin We know how to find (or pproximte) sin given, ut how do we reverse the proess? How do we find given sin? First, we note tht the solution to the prolem n e written symolilly s either rsin or rsin nd sin oth represent the sme thing. sin Both expressions re red is the ngle whose sine is

4 382 5 TRIGONOMETRIC FUNCTIONS CAUTION It is importnt to note tht sin does not men /(sin 0.496). The supersript is prt of funtion symol, nd sin represents the inverse sine funtion. Inverse trigonometri funtions re developed in detil in Setion 5-9. Fortuntely, we n find diretly using lultor. Most lultors of the type used in this ook hve the funtion keys sin, os, nd tn or their equivlents (hek your mnul). These funtion keys tke us from trigonometri rtio k to the orresponding ute ngle in degree mesure when the lultor is in degree mode. Thus, if sin 0.496, then we n write either rsin or sin We hoose the ltter nd proeed s follows: sin or To the nerest hundredth degree To the nerest minute Chek sin Explore/Disuss 2 Solve eh of the following for to the nerest hundredth of degree using lultor. Explin why n error messge ours in one of the prolems. (A) os (B) tn.438 (C) sin.438 EXAMPLE 2 Right Tringle Solution Solve the right tringle with 4.32 entimeters nd 2.62 entimeters. Compute the ngle mesures to the nerest 0. Solution Drw figure nd lel the known prts (Fig. 3): FIGURE m 4.32 m Solve for tn tn or [(0.2)(60)] 2 0 to nerest 0

5 Solve for Solving Right Tringles 383 Solve for sin 2.62 Or use s entimeters sin 3.2 or, using the Pythgoren theorem, entimeters Note the slight differene in the vlues otined for (5.05 versus 5.06). This ws used y rounding to the nerest 0 in the first lultion for. MATCHED PROBLEM 2 Solve the right tringle with.38 kilometers nd 6.73 kilometers. EXAMPLE 3 Solution FIGURE 4 Geometry If pentgon ( five-sided regulr polygon) is insried in irle of rdius 5.35 entimeters, find the length of one side of the pentgon. Sketh figure nd insert tringle ACB with C t the enter (Fig. 4). Add the uxiliry line CD s indited. We will find AD nd doule it to find the length of the side wnted. C B D 5.35 A Angle ACB Angle ACD sin (ngle ACD) AD AC sin 36 Ext Ext AD AC sin (ngle ACD) 3.4 entimeters AB 2AD 6.28 entimeters MATCHED PROBLEM 3 If squre of side 43.6 meters is insried in irle, wht is the rdius of the irle?

6 384 5 TRIGONOMETRIC FUNCTIONS EXAMPLE 4 FIGURE 5 Arhiteture In designing house n rhitet wishes to determine the mount of overhng of roof so tht it shdes the entire south wll t noon during the summer solstie (Fig. 5). Minimlly, how muh overhng should e provided for this purpose? Winter solstie sun (noon) Summer solstie sun (noon) 8 ft 32 Solution FIGURE 6 From the figure we drw the following relted right tringle (Fig. 6) nd solve for x: 8 x tn x x tn 9.7 feet MATCHED PROBLEM 4 With the overhng found in Exmple 4, how fr will the shdow of the overhng ome down the wll t noon during the winter solstie? Answers to Mthed Prolems. 42.2, 20.2 m, 8.3 m 2. 40, 78 20, 6.87 km m 4.. ft

7 5-5 Solving Right Tringles 385 EXERCISE 5-5 A For the tringle in the figure for Prolems 2, write the rtios of sides orresponding to eh trigonometri funtion given in Prolems 6. Do not look k t the definitions.. sin 2. ot 3. s 4. os 5. tn 6. se Prolems 3 36 give geometri interprettion of the trigonometri rtios. Refer to the figure, where 0 is the enter of irle of rdius, is the ute ngle AOD, D is the intersetion point of the terminl side of ngle with the irle, nd EC is tngent to the irle t D. E F ot D s sin tn Eh rtio in Prolems 7 2 defines trigonometri funtion of (refer to the figure for Prolems 2). Indite whih funtion without looking k t the definitions. 7. / 8. / 9. / 0. /. / 2. / In Prolems 3 8, find eh ute ngle in degree mesure to two deiml ples using lultor. 3. os sin tn os sin tn.993 B Solve eh tringle in Prolems 9 30 using the informtion given nd the tringle leling in the figure for Prolems O os A se 3. Explin why (A) os OA (B) ot DE (C) se OC 32. Explin why (A) sin AD (B) tn DC (C) s OE 33. Explin wht hppens to eh of the following s the ute ngle pprohes 90. (A) os (B) ot (C) se 34. Explin wht hppens to eh of the following s the ute ngle pprohes 90. (A) sin (B) tn (C) s 35. Explin wht hppens to eh of the following s the ute ngle pprohes 0. (A) sin (B) tn (C) s 36. Explin wht hppens to eh of the following s the ute ngle pprohes 0. (A) os (B) ot (C) se B C , , 22.4 C , , , , , , , , Show tht d h ot ot h , , 65 d

8 386 5 TRIGONOMETRIC FUNCTIONS 38. Show tht d h ot ot d h time, no timing devies existed to mesure the veloity of free-flling ody, so Glileo used the inlined plne to slow the motion down.) A steel ll is rolled down glss plne inlined t 8.0. Approximte g to one deiml ple if t the end of 3.0 seonds the ll hs mesured veloity of 4.2 meters per seond. APPLICATIONS 39. Surveying. Find the height of tree (growing on level ground) if t point 05 feet from the se of the tree the ngle to its top reltive to the horizontl is found to e Air Sfety. To mesure the height of loud eiling over n irport, serhlight is direted stright upwrd to produe lighted spot on the louds. Five hundred meters wy n oserver reports the ngle of the spot reltive to the horizontl to e How high (to the nerest meter) re the louds ove the irport? 4. Engineering. If trin lims t onstnt ngle of 23, how mny vertil feet hs it limed fter going mile? ( mile 5,280 feet) 42. Air Sfety. If jet irliner lims t n ngle of 5 30 with onstnt speed of 35 miles per hour, how long will it tke (to the nerest minute) to reh n ltitude of 8.00 miles? Assume there is no wind. 43. Astronomy. Find the dimeter of the moon (to the nerest mile) if t 239,000 miles from Erth it produes n ngle of 32 reltive to n oserver on Erth. 44. Astronomy. If the sun is 93,000,000 miles from Erth nd its dimeter is opposite n ngle of 32 reltive to n oserver on Erth, wht is the dimeter of the sun (to two signifint digits)? 45. Geometry. If irle of rdius 4 entimeters hs hord of length 3 entimeters, find the entrl ngle tht is opposite this hord (to the nerest degree). 46. Geometry. Find the length of one side of nine-sided regulr polygon insried in irle of rdius 4.06 inhes. 47. Physis. In ourse in physis it is shown tht the veloity v of ll rolling down n inlined plne (negleting ir resistne nd frition) is given y v gt sin where g is grvittionl onstnt (elertion due to grvity), t is time, nd is the ngle of inlintion of the plne (see the figure). Glileo ( ) used this eqution in the form g v t sin to estimte g fter mesuring v experimentlly. (At tht 48. Physis. Refer to Prolem 47. A steel ll is rolled down glss plne inlined t 4.0. Approximte g to one deiml ple if t the end of 4.0 seonds the ll hs mesured veloity of 9.0 feet per seond. 49. Engineering Cost Anlysis. A le television ompny wishes to run le from ity to resort islnd 3 miles offshore. The le is to go long the shore, then to the islnd underwter, s indited in the ompnying figure. The ost of running the le long the shore is \$5,000 per mile nd underwter, \$25,000 per mile. City 20 miles Resort Islnd 3 miles Shore (A) Referring to the figure, show tht the ost in terms of is given y C() 75,000 se 45,000 tn 300,000 (B) Clulte tle of osts, eh ost to the nerest dollr, for the following vlues of : 0, 20, 30, 40, nd 50. (Notie how the osts vry with. In ourse in lulus, students re sked to find so tht the ost is minimized.) 50. Engineering Cost Anlysis. Refer to Prolem 49. Suppose the islnd is 4 miles offshore nd the ost of running the le long the shore is \$20,000 per mile nd underwter, \$30,000 per mile. (A) Referring to the figure for Prolem 49 with pproprite hnges, show tht the ost in terms of is given y C() 20,000 se 80,000 tn 400,000 (B) Clulte tle of osts, eh ost to the nerest dollr, for the following vlues of : 0, 20, 30, 40, nd 50.

9 5-6 Grphing Bsi Trigonometri Funtions Geometry. Find r in the ompnying figure (to two signifint digits) so tht the irle is tngent to ll three sides of the isoseles tringle. [Hint: The rdius of irle is perpendiulr to tngent line t the point of tngeny.] 52. Geometry. Find r in the ompnying figure (to two signifint digits) so tht the smller irle is tngent to the lrger irle nd the two sides of the ngle. [See the hint in Prolem 5.] 30 r 2.0 meters 30 r 2.0 in. Setion 5-6 Grphing Bsi Trigonometri Funtions Periodi Funtions Grphs of y sin x nd y os x Grphs of y tn x nd y ot x Grphs of y s x nd y se x Grphs on Grphing Utility Consider the grphs of sunrise times nd sound wves shown in Figure. Wht is ommon feture of the two grphs? Both represent repetitive phenomen; tht is, oth pper to e periodi. Trigonometri funtions re prtiulrly suited to desrie periodi phenomen. Sunrise time F A J A O D F A J A O D Pressure t erdrum Month Sunrise times nd time of yer () FIGURE Periodi phenomen. Time (seonds) Soundwve rriving t erdrum () In this setion we disuss the grphs of the six trigonometri funtions with rel numer domins introdued erlier. We lso disuss the domins, rnges, nd periodi properties of these funtions. The irulr funtions introdued in Setion 5-2 will prove prtiulrly useful in this regrd. It ppers there is lot to rememer in this setion. However, you only need to e fmilir with the grphs nd properties of the sine, osine, nd tngent funtions. The reiprol reltionships disussed erlier will enle you to determine the grphs nd properties of the other three trigonometri funtions from the grphs nd properties of the sine, osine, nd tngent funtions.

### SECTION 7-2 Law of Cosines

516 7 Additionl Topis in Trigonometry h d sin s () tn h h d 50. Surveying. The lyout in the figure t right is used to determine n inessile height h when seline d in plne perpendiulr to h n e estlished

### Lesson 18.2: Right Triangle Trigonometry

Lesson 8.: Right Tringle Trigonometry lthough Trigonometry is used to solve mny prolems, historilly it ws first pplied to prolems tht involve right tringle. This n e extended to non-right tringles (hpter

### Right Triangle Trigonometry 8.7

304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R1 8.7 Right Tringle Trigonometry R1 8.7 Right Tringle Trigonometry T E G T I N G S T R T E D The origins of trigonometry, from the Greek trigonon (ngle) nd

### The area of the larger square is: IF it s a right triangle, THEN + =

8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht

### Ratio and Proportion

Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty

### Right Triangle Trigonometry

CONDENSED LESSON 1.1 Right Tringle Trigonometr In this lesson ou will lern out the trigonometri rtios ssoited with right tringle use trigonometri rtios to find unknown side lengths in right tringle use

### Essential Question What are the Law of Sines and the Law of Cosines?

9.7 TEXS ESSENTIL KNOWLEDGE ND SKILLS G.6.D Lw of Sines nd Lw of osines Essentil Question Wht re the Lw of Sines nd the Lw of osines? Disovering the Lw of Sines Work with prtner.. opy nd omplete the tle

### 8.2 Trigonometric Ratios

8.2 Trigonometri Rtios Ojetives: G.SRT.6: Understnd tht y similrity, side rtios in right tringles re properties of the ngles in the tringle, leding to definitions of trigonometri rtios for ute ngles. For

### Section 5-4 Trigonometric Functions

5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form

### 2.1 ANGLES AND THEIR MEASURE. y I

.1 ANGLES AND THEIR MEASURE Given two interseting lines or line segments, the mount of rottion out the point of intersetion (the vertex) required to ring one into orrespondene with the other is lled the

### D e c i m a l s DECIMALS.

D e i m l s DECIMALS www.mthletis.om.u Deimls DECIMALS A deiml numer is sed on ple vlue. 214.84 hs 2 hundreds, 1 ten, 4 units, 8 tenths nd 4 hundredths. Sometimes different 'levels' of ple vlue re needed

### Geometry 7-1 Geometric Mean and the Pythagorean Theorem

Geometry 7-1 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the

### PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions

PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * hllenge questions e The ll will strike the ground 1.0 s fter it is struk. Then v x = 20 m s 1 nd v y = 0 + (9.8 m s 2 )(1.0 s) = 9.8 m s 1 The speed

### 8. Hyperbolic triangles

8. Hyperoli tringles Note: This yer, I m not doing this mteril, prt from Pythgors theorem, in the letures (nd, s suh, the reminder isn t exminle). I ve left the mteril s Leture 8 so tht (i) nyody interested

### Three squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names.

1- The Pythgoren Theorem MAIN IDEA Find length using the Pythgoren Theorem. New Voulry leg hypotenuse Pythgoren Theorem Mth Online glenoe.om Extr Exmples Personl Tutor Self-Chek Quiz Three squres with

### Right Triangle Trigonometry for College Algebra

Right Tringle Trigonometry for ollege Alger B A sin os A = = djent A = = tn A = = djent sin B = = djent os B = = tn B = = djent ontents I. Bkground nd Definitions (exerises on pges 3-4) II. The Trigonometri

### The remaining two sides of the right triangle are called the legs of the right triangle.

10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right

### excenters and excircles

21 onurrene IIi 2 lesson 21 exenters nd exirles In the first lesson on onurrene, we sw tht the isetors of the interior ngles of tringle onur t the inenter. If you did the exerise in the lst lesson deling

### THE PYTHAGOREAN THEOREM

THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this

### 8-1. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

8-1 The Pythgoren Theorem nd Its Converse Voulry Review 1. Write the squre nd the positive squre root of eh numer. Numer Squre Positive Squre Root 9 81 3 1 4 1 16 1 2 Voulry Builder leg (noun) leg Relted

### Introduction. Law of Cosines. a 2 b2 c 2 2bc cos A. b2 a 2 c 2 2ac cos B. c 2 a 2 b2 2ab cos C. Example 1

3330_060.qxd 1/5/05 10:41 M Pge 439 Setion 6. 6. Lw of osines 439 Lw of osines Wht you should lern Use the Lw of osines to solve olique tringles (SSS or SS). Use the Lw of osines to model nd solve rel-life

### Proving the Pythagorean Theorem

Proving the Pythgoren Theorem Proposition 47 of Book I of Eulid s Elements is the most fmous of ll Eulid s propositions. Disovered long efore Eulid, the Pythgoren Theorem is known y every high shool geometry

### Functions A B C D E F G H I J K L. Contents:

Funtions Contents: A reltion is n set of points whih onnet two vriles. A funtion, sometimes lled mpping, is reltion in whih no two different ordered pirs hve the sme -oordinte or first omponent. Algeri

### Thank you for participating in Teach It First!

Thnk you for prtiipting in Teh It First! This Teh It First Kit ontins Common Core Coh, Mthemtis teher lesson followed y the orresponding student lesson. We re onfident tht using this lesson will help you

### Simple Electric Circuits

Simple Eletri Ciruits Gol: To uild nd oserve the opertion of simple eletri iruits nd to lern mesurement methods for eletri urrent nd voltge using mmeters nd voltmeters. L Preprtion Eletri hrges move through

### 10.3 Systems of Linear Equations: Determinants

758 CHAPTER 10 Systems of Equtions nd Inequlities 10.3 Systems of Liner Equtions: Determinnts OBJECTIVES 1 Evlute 2 y 2 Determinnts 2 Use Crmer s Rule to Solve System of Two Equtions Contining Two Vriles

### The Pythagorean Theorem

The Pythgoren Theorem Pythgors ws Greek mthemtiin nd philosopher, orn on the islnd of Smos (. 58 BC). He founded numer of shools, one in prtiulr in town in southern Itly lled Crotone, whose memers eventully

### 11.1 Conic sections (conics)

. Coni setions onis Coni setions re formed the intersetion of plne with right irulr one. The tpe of the urve depends on the ngle t whih the plne intersets the surfe A irle ws studied in lger in se.. We

### It may be helpful to review some right triangle trigonometry. Given the right triangle: C = 90º

Ryn Lenet Pge 1 Chemistry 511 Experiment: The Hydrogen Emission Spetrum Introdution When we view white light through diffrtion grting, we n see ll of the omponents of the visible spetr. (ROYGBIV) The diffrtion

### 5.6 The Law of Cosines

44 HPTER 5 nlyti Trigonometry 5.6 The Lw of osines Wht you ll lern out Deriving the Lw of osines Solving Tringles (SS, SSS) Tringle re nd Heron s Formul pplitions... nd why The Lw of osines is n importnt

### Angles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example

2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel

### 1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.

. Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry

### 11. PYTHAGORAS THEOREM

11. PYTHAGORAS THEOREM 11-1 Along the Nile 2 11-2 Proofs of Pythgors theorem 3 11-3 Finding sides nd ngles 5 11-4 Semiirles 7 11-5 Surds 8 11-6 Chlking hndll ourt 9 11-7 Pythgors prolems 10 11-8 Designing

### Radius of the Earth - Radii Used in Geodesy James R. Clynch Naval Postgraduate School, 2002

dius of the Erth - dii Used in Geodesy Jmes. Clynh vl Postgrdute Shool, 00 I. Three dii of Erth nd Their Use There re three rdii tht ome into use in geodesy. These re funtion of ltitude in the ellipsoidl

### Lesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle

: Using Trigonometry to Find Side Lengths of n Aute Tringle Clsswork Opening Exerise. Find the lengths of d nd e.. Find the lengths of x nd y. How is this different from prt ()? Exmple 1 A surveyor needs

### Practice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn

Prtie Test 2 1. A highwy urve hs rdius of 0.14 km nd is unnked. A r weighing 12 kn goes round the urve t speed of 24 m/s without slipping. Wht is the mgnitude of the horizontl fore of the rod on the r?

### Know the sum of angles at a point, on a straight line and in a triangle

2.1 ngle sums Know the sum of ngles t point, on stright line n in tringle Key wors ngle egree ngle sum n ngle is mesure of turn. ngles re usully mesure in egrees, or for short. ngles tht meet t point mke

### Chapter. Contents: A Constructing decimal numbers

Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting

### SOLVING EQUATIONS BY FACTORING

316 (5-60) Chpter 5 Exponents nd Polynomils 5.9 SOLVING EQUATIONS BY FACTORING In this setion The Zero Ftor Property Applitions helpful hint Note tht the zero ftor property is our seond exmple of getting

### Calculating Principal Strains using a Rectangular Strain Gage Rosette

Clulting Prinipl Strins using Retngulr Strin Gge Rosette Strin gge rosettes re used often in engineering prtie to determine strin sttes t speifi points on struture. Figure illustrtes three ommonly used

### 4.5 The Converse of the

Pge 1 of. The onverse of the Pythgoren Theorem Gol Use the onverse of Pythgoren Theorem. Use side lengths to lssify tringles. Key Words onverse p. 13 grdener n use the onverse of the Pythgoren Theorem

### Lesson 18.3: Triangle Trigonometry ( ) : OBTUSE ANGLES

Lesson 1.3: Tringle Trigonometry We now extend te teory of rigt tringle trigonometry to non-rigt or olique tringles. Of te six omponents wi form tringle, tree sides nd tree ngles, te possiilities for omintion

### Words Symbols Diagram. abcde. a + b + c + d + e

Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To

### Sine and Cosine Ratios. For each triangle, find (a) the length of the leg opposite lb and (b) the length of the leg adjacent to lb.

- Wht You ll ern o use sine nd osine to determine side lengths in tringles... nd Wh o use the sine rtio to estimte stronomil distnes indiretl, s in Emple Sine nd osine tios hek Skills You ll Need for Help

### Lesson 12.1 Trigonometric Ratios

Lesson 12.1 rigonometric Rtios Nme eriod Dte In Eercises 1 6, give ech nswer s frction in terms of p, q, nd r. 1. sin 2. cos 3. tn 4. sin Q 5. cos Q 6. tn Q p In Eercises 7 12, give ech nswer s deciml

### The Pythagorean Theorem Tile Set

The Pythgoren Theorem Tile Set Guide & Ativities Creted y Drin Beigie Didx Edution 395 Min Street Rowley, MA 01969 www.didx.om DIDAX 201 #211503 1. Introdution The Pythgoren Theorem sttes tht in right

### Example

6. SOLVING RIGHT TRINGLES In the right tringle B shwn in Figure 6.1, the ngles re dented y α t vertex, β t vertex B, nd t vertex. The lengths f the sides ppsite the ngles α, β, nd re dented y,, nd. Nte

### Final Exam covers: Homework 0 9, Activities 1 20 and GSP 1 6 with an emphasis on the material covered after the midterm exam.

MTH 494.594 / FINL EXM REVIEW Finl Exm overs: Homework 0 9, tivities 1 0 nd GSP 1 6 with n emphsis on the mteril overed fter the midterm exm. You my use oth sides of one 3 5 rd of notes on the exm onepts

### 15. Let f (x) = 3x Suppose rx 2 + sx + t = 0 where r 0. Then x = 24. Solve 5x 25 < 20 for x. 26. Let y = 7x

Pretest Review The pretest will onsist of 0 problems, eh of whih is similr to one of the following 49 problems If you n do problems like these 49 listed below, you will hve no problem with the pretest

### OVERVIEW Prove & Use the Laws of Sines & Cosines G.SRT.10-HONORS

OVERVIEW Prove & Use te Lws of Sines & osines G.SRT.10-HONORS G.SRT.10 (HONORS ONLY) Prove te Lws of Sines nd osines nd use tem to solve prolems. No interprettion needed - prove te Lw of Sines nd te Lw

### 11.2 The Law of Sines

894 Applitions of Trigonometry 11. The Lw of Sines Trigonometry literlly mens mesuring tringles nd with Chpter 10 under our belts, we re more thn prepred to do just tht. The min gol of this setion nd the

### State the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127

ngles 2 CHTER 2.1 Tringles Drw tringle on pper nd lel its ngles, nd. Ter off its orners. Fit ngles, nd together. They mke stright line. This shows tht the ngles in this tringle dd up to 180 ut it is not

### The theorem of. Pythagoras. Opening problem

The theorem of 8 Pythgors ontents: Pythgors theorem [4.6] The onverse of Pythgors theorem [4.6] Prolem solving [4.6] D irle prolems [4.6, 4.7] E Three-dimensionl prolems [4.6] Opening prolem The Louvre

### Angles and Triangles

nges nd Tringes n nge is formed when two rys hve ommon strting point or vertex. The mesure of n nge is given in degrees, with ompete revoution representing 360 degrees. Some fmiir nges inude nother fmiir

### MBF 3C Unit 2 Trigonometry Outline

Dy MF 3 Unit 2 Trigonometry Outline Lesson Title Speifi Expettions 1 Review Trigonometry Solving for Sides Review Gr. 10 2 Review Trigonometry Solving for ngles Review Gr. 10 3 Trigonometry in the Rel

### SOLVING QUADRATIC EQUATIONS BY FACTORING

6.6 Solving Qudrti Equtions y Ftoring (6 31) 307 In this setion The Zero Ftor Property Applitions 6.6 SOLVING QUADRATIC EQUATIONS BY FACTORING The tehniques of ftoring n e used to solve equtions involving

### Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.

Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd

### Pythagoras theorem is one of the most popular theorems. Paper Folding And The Theorem of Pythagoras. Visual Connect in Teaching.

in the lssroom Visul Connet in Tehing Pper Folding And The Theorem of Pythgors Cn unfolding pper ot revel proof of Pythgors theorem? Does mking squre within squre e nything more thn n exerise in geometry

### MATH PLACEMENT REVIEW GUIDE

MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your

### How to Graphically Interpret the Complex Roots of a Quadratic Equation

Universit of Nersk - Linoln DigitlCommons@Universit of Nersk - Linoln MAT Em Epositor Ppers Mth in the Middle Institute Prtnership 7-007 How to Grphill Interpret the Comple Roots of Qudrti Eqution Crmen

### End of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.

End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte Complete the missing numers in the sequenes elow. 8 30 3 28 2 9 25 00 75 25 2 Put irle round ll of the following shpes whih hve 3 shded.

### Two special Right-triangles 1. The

Mth Right Tringle Trigonometry Hndout B (length of ) - c - (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Right-tringles. The

### Consumption and investment spending. Cambridge University Press 2012 Economics for the IB Diploma 1

Supplementry mteril for Chpter 9 9.8 Understnding ggregte demnd nd the multiplier in terms of the Keynesin ross model (supplementry mteril, reommended for higher level) This mteril is inluded for the interested

### Sect 8.3 Triangles and Hexagons

13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed two-dimensionl geometric figure consisting of t lest three line segments for its

### Problem Set 2 Solutions

University of Cliforni, Berkeley Spring 2012 EE 42/100 Prof. A. Niknej Prolem Set 2 Solutions Plese note tht these re merely suggeste solutions. Mny of these prolems n e pprohe in ifferent wys. 1. In prolems

### Chapter15 SAMPLE. Simultaneous equations. Contents: A B C D. Graphical solution Solution by substitution Solution by elimination Problem solving

Chpter15 Simultneous equtions Contents: A B C D Grphil solution Solution y sustitution Solution y elimintion Prolem solving 308 SIMULTANEOUS EQUATIONS (Chpter 15) Opening prolem Ewen wnts to uy pie, ut

### Right-angled triangles

13 13A Pythgors theorem 13B Clulting trigonometri rtios 13C Finding n unknown side 13D Finding ngles 13E Angles of elevtion nd depression Right-ngled tringles Syllus referene Mesurement 4 Right-ngled tringles

### CHAPTER 4: POLYGONS AND SOLIDS. 3 Which of the following are regular polygons? 4 Draw a pentagon with equal sides but with unequal angles.

Mthemtis for Austrli Yer 6 - Homework POLYGONS AND SOLIDS (Chpter 4) CHAPTER 4: POLYGONS AND SOLIDS 4A POLYGONS 3 Whih of the following re regulr polygons? A polygon is lose figure whih hs only stright

### Fractions: Arithmetic Review

Frtions: Arithmeti Review Frtions n e interprete s rtios omprisons of two quntities. For given numer expresse in frtion nottion suh s we ll the numertor n the enomintor n it is helpful to interpret this

### Reasoning to Solve Equations and Inequalities

Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing

### Pythagoras theorem and trigonometry (2)

HPTR 10 Pythgors theorem nd trigonometry (2) 31 HPTR Liner equtions In hpter 19, Pythgors theorem nd trigonometry were used to find the lengths of sides nd the sizes of ngles in right-ngled tringles. These

### Chapter 10 Geometry: Angles, Triangles and Distance

hpter 10 Geometry: ngles, Tringles nd Distne In setion 1 we egin y gthering together fts out ngles nd tringles tht hve lredy een disussed in previous grdes. This time the ide is to se student understnding

### SAMPLE. Trigonometric ratios and applications

jetives H P T E R 12 Trigonometri rtios nd pplitions To solve prtil prolems using the trigonometri rtios To use the sine rule nd the osine rule to solve prolems To find the re of tringle given two sides

### STRAND I: Geometry and Trigonometry. UNIT I2 Trigonometric Problems: Text * * Contents. Section. I2.1 Mixed Problems Using Trigonometry

Mthemtics SKE: STRND I UNIT I Trigonometric Prolems: Text STRND I: Geometry nd Trigonometry I Trigonometric Prolems Text ontents Section * * * I. Mixed Prolems Using Trigonometry I. Sine nd osine Rules

### 1. Area under a curve region bounded by the given function, vertical lines and the x axis.

Ares y Integrtion. Are uner urve region oune y the given funtion, vertil lines n the is.. Are uner urve region oune y the given funtion, horizontl lines n the y is.. Are etween urves efine y two given

### Basic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }

ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All

### ASYMPTOTES HORIZONTAL ASYMPTOTES VERTICAL ASYMPTOTES. An asymptote is a line which a function gets closer and closer to but never quite reaches.

UNFAMILIAR FUNCTIONS (Chpter 19) 547 B ASYMPTOTES An smptote is line whih funtion gets loser n loser to but never quite rehes. In this ourse we onsier smptotes whih re horizontl or vertil. HORIZONTAL ASYMPTOTES

### Simple Nonlinear Graphs

Simple Nonliner Grphs Curriulum Re www.mthletis.om Simple SIMPLE Nonliner NONLINEAR Grphs GRAPHS Liner equtions hve the form = m+ where the power of (n ) is lws. The re lle Liner euse their grphs re stright

### Mathematics in Art and Architecture GEK1518K

Mthemtics in Art nd Architecture GEK1518K Helmer Aslksen Deprtment of Mthemtics Ntionl University of Singpore slksen@mth.nus.edu.sg www.mth.nus.edu.sg/slksen/ The Golden Rtio The Golden Rtio Suppose we

### Maximum area of polygon

Mimum re of polygon Suppose I give you n stiks. They might e of ifferent lengths, or the sme length, or some the sme s others, et. Now there re lots of polygons you n form with those stiks. Your jo is

### Square Roots Teacher Notes

Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this

### Line Integrals and Vector Fields

Line Integrls nd Vetor Fields The origin of the notion of line integrl (relly pth integrl) omes from the physil notion of work. We will see tht prtiulr pplition presently. Our first tsk is to give definition

### PYTHAGORAS THEOREM. Answers. Edexcel GCSE Mathematics (Linear) 1MA0

Edexel GSE Mthemtis (Liner) 1M0 nswers PYTHGORS THEOREM Mterils required for exmintion Ruler grduted in entimetres nd millimetres, protrtor, ompsses, pen, H penil, erser. Tring pper my e used. Items inluded

### Heron s Formula for Triangular Area

Heron s Formul for Tringulr Are y Christy Willims, Crystl Holom, nd Kyl Gifford Heron of Alexndri Physiist, mthemtiin, nd engineer Tught t the museum in Alexndri Interests were more prtil (mehnis, engineering,

### Name: Lab Partner: Section:

Chpter 4 Newton s 2 nd Lw Nme: Lb Prtner: Section: 4.1 Purpose In this experiment, Newton s 2 nd lw will be investigted. 4.2 Introduction How does n object chnge its motion when force is pplied? A force

### The AVL Tree Rotations Tutorial

The AVL Tree Rottions Tutoril By John Hrgrove Version 1.0.1, Updted Mr-22-2007 Astrt I wrote this doument in n effort to over wht I onsider to e drk re of the AVL Tree onept. When presented with the tsk

### Muscle and Joint Forces

Musle nd Joint Fores Ozky nd Nordin Chpters 4 nd 5 (pge 49-115) Leture outline omponents of joint fore pplitions of stti nlysis: mehnis of the elbow during lifting mehnis of the shoulder during lifting

0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions

### Graphs on Logarithmic and Semilogarithmic Paper

0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl

### 9.1 PYTHAGOREAN THEOREM (right triangles)

Simplifying Rdicls: ) 1 b) 60 c) 11 d) 3 e) 7 Solve: ) x 4 9 b) 16 80 c) 9 16 9.1 PYTHAGOREAN THEOREM (right tringles) c If tringle is right tringle then b, b re the legs * c is clled the hypotenuse (side

### PYTHAGORAS THEOREM 8YEARS. A guide for teachers - Years 8 9. The Improving Mathematics Education in Schools (TIMES) Project

The Improving Mthemtis Edution in Shools (TIMES) Projet PYTHGORS THEOREM guide for tehers - Yers 8 9 MESUREMENT ND GEOMETRY Module 15 June 2011 8YERS 9 Pythgors theorem (Mesurement nd Geometry: Module

### Geometry Notes SIMILAR TRIANGLES

Similr Tringles Pge 1 of 6 SIMILAR TRIANGLES Objectives: After completing this section, you shoul be ble to o the following: Clculte the lengths of sies of similr tringles. Solve wor problems involving

### Interior and exterior angles add up to 180. Level 5 exterior angle

22 ngles n proof Ientify interior n exterior ngles in tringles n qurilterls lulte interior n exterior ngles of tringles n qurilterls Unerstn the ie of proof Reognise the ifferene etween onventions, eﬁnitions

### Net Change and Displacement

mth 11, pplictions motion: velocity nd net chnge 1 Net Chnge nd Displcement We hve seen tht the definite integrl f (x) dx mesures the net re under the curve y f (x) on the intervl [, b] Any prt of the

### PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1

PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.

### Geometry of Crystals. Crystal is a solid composed of atoms, ions or molecules that demonstrate long range periodic order in three dimensions

Geometry of Crystls Crystl is solid omposed of toms, ions or moleules tht demonstrte long rnge periodi order in three dimensions The Crystlline Stte Stte of Mtter Fixed Volume Fixed Shpe Order Properties

### Homework Assignment 1 Solutions

Dept. of Mth. Sci., WPI MA 1034 Anlysis 4 Bogdn Doytchinov, Term D01 Homework Assignment 1 Solutions 1. Find n eqution of sphere tht hs center t the point (5, 3, 6) nd touches the yz-plne. Solution. The

### Let us recall some facts you have learnt in previous grades under the topic Area.

6 Are By studying this lesson you will be ble to find the res of sectors of circles, solve problems relted to the res of compound plne figures contining sectors of circles. Ares of plne figures Let us
{"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}