# 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.

Save this PDF as:

Size: px
Start display at page:

Download "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."

## Transcription

1 - 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 For eh tringle, find () the length of the leg l nd () the length of the leg djent to l.. 9; ; " New Voulr sine osine identit 9 0; "9 esson Pln Ojetives o use sine nd osine to determine side lengths in tringles Emples Writing Sine nd osine tios el-world onnetion Using the Inverse of osine nd Sine Mth kground Using Sine nd osine in ringles he tngent rtio, s ou hve seen, involves oth legs of right tringle. he sine nd osine rtios involve one leg nd the hpotenuse. sine of & osine of & leg / hpotenuse leg djent to / hpotenuse Hpotenuse eg djent to eg unit irle hs rdius nd enter (0,0) in the oordinte plne. For ll rel vlues of u, the point tht is rehed trveling u rdins from point (,0) in ounterlokwise diretion hs oordintes (os u, sin u). More Mth kground: p. D el-world onnetion For n ngle of given sie, the sine nd osine rtios re onstnt, no mtter where the ngle is loted. Quik hek hese equtions n e revited: djent sin hpotenuse os hpotenuse EXMPE Writing Sine nd osine tios Use the tringle to write eh rtio.. sin sin hpotenuse. os djent os hpotenuse. sin sin hpotenuse d. os djent os hpotenuse. Write the sine nd osine rtios for &X nd &Y. See right.. ritil hinking When does sin X os Y Eplin. sin X os Y when lx nd ly re omplementr. X Z 0 6 esson - Sine nd osine tios 9 Y 6. sin X 0 ; os X 0 ; sin Y 0 ; os Y 6 0 esson Plnning nd esoures See p. E for list of the resoures tht support this lesson. ell inger Prtie hek Skills You ll Need For intervention, diret students to: Writing ngent tios esson -: Emple Etr Skills, Word Prolems, Proof Prtie, h. Speil Needs Present the mnemoni devie SOHHO for the definition of the three trigonometri rtios: Sine is Opposite over Hpotenuse; osine is djent over Hpotenuse; ngent is Opposite over djent. lerning stle: verl elow evel Hve students drw nd mesure right tringles to mke tle of sine nd osine vlues for the ngles in the set {0, 0,..., 0 }. lerning stle: visul 9

2 . eh uided Instrution Visul erners Hve students displ poster listing the trigonometri rtios. dditionl Emples Use the tringle to find sin, os, sin, nd os. 6 sin 0, os 0, 6 sin 0, os 0 0-ft wire supporting flgpole forms ngle with the flgpole. o the nerest foot, how high is the flgpole 0 ft 0 6 For: Sine nd osine tivit Use: Intertive etook, - Quik hek One w to desrie the reltionship of sine nd osine is to s tht sin os (90 - ) for vlues of etween 0 nd 90. his tpe of eqution is lled n identit euse it is true for ll the llowed vlues of the vrile. You will disover other identities in the eerises. EXMPE el-world onnetion stronom he trigonometri rtios hve een known for enturies peoples in mn ultures. he Polish stronomer Niolus opernius (-) developed method for determining the sies of orits of plnets loser to the sun thn Erth. he ke to his method ws determining when the plnets were in the position shown in the digrm, nd then mesuring the ngle to find. If. for Merur, how fr is Merur from the sun in stronomil units (U) One stronomil unit is defined s the verge distne from Erth to the enter of the sun, out 9 million miles. sin. Use the sine rtio. sin. Solve for Use lultor. Merur is out 0. U from the sun.. If 6 for Venus, how fr is Venus from the sun in U. out how mn miles from the sun is Venus Merur 66,960,000 mi;,0,000 mi When ou know the leg nd hpotenuse lengths of right tringle, ou n use inverse of sine nd inverse of osine to find the mesures of the ute ngles. EXMPE Erth Using the Inverse of osine nd Sine U Sun not to sle out 0. U 6 ft right tringle hs leg. units long nd hpotenuse.0 units long. Find the mesures of its ute ngles to the nerest degree., 6 esoures Dil Notetking uide - Dil Notetking uide - dpted Instrution losure right tringle whose hpotenuse is m long ontins 6 ngle. Find the lengths of its legs to one deiml ple. 6. m,.6 m Prolem Solving Hint hink of os -. (.0) s the ngle whose. osine is.0, nd sin -. (.0) s the ngle whose sine is. the quotient.0. Quik hek Find m& to the nerest degree. 0 hpter ight ringles nd rigonometr. F. O Method Method os..0 Find the trigonometri rtio. sin..0 m& os -Q..0 Use the inverse. m& sin - Q..0 OS -..0 Use lultor. SIN m& < m& < Find the vlue of. ound our nswer to the nerest degree dvned erners Enourge students to mke onjetures out the vlues of sin 0, os 0, sin 90, nd os 90, defend their onjetures, nd then hek the vlues on lultor. lerning stle: verl English nguge erners E Help students distinguish etween sine nd inverse sine. he sine of n ngle is rtio, or numer. he inverse sine of numer, or rtio, is n ngle mesure. So, inverses re used to find ngle mesures. lerning stle: verl

3 EXEISES For more eerises, see Etr Skill, Word Prolem, nd Proof Prtie. Prtie nd Prolem Solving Prtie Emple for Help Emple (pge 9) Emple (pge 0) Emple (pge 0) Write the rtios for sin M nd os M... M M. 9 " ; 9 ; 9 Find the vlue of. ound nswers to the nerest tenth Esltors n esltor in the suw sstem of St. Petersurg, ussi, hs vertil rise of 9 ft 9. in., nd rises t n ngle of 0.. How long is the esltor ound our nswer to the nerest foot. 0 ft Find the vlue of. ound nswers to the nerest degree ; " M. Prtie ssignment uide -0 hllenge -6 est Prep -0 Mied eview - Homework Quik hek o hek students understnding of ke skills nd onepts, go over Eerises,,,,. Error Prevention! Eerises 6, Some students m need help solving equtions with the vrile in the denomintor. eview tehniques suh s rossmultiplition nd tking the reiprol of eh side. Eerise ell students tht there is lso otngent rtio. sk: Wht do ou think is the otngent rtio djent ppl Your Skills. onstrution rlos is plnning to uild grin in with rdius of ft. He reds tht the reommended slnt of the roof is. He wnts the roof to overhng the edge of the in ft. Wht should the length e ive our nswer in feet nd inhes. out ft in. ft overhng PS Enrihment uided Prolem Solving etehing dpted Prtie Prtie Nme lss Dte Use wht ou know out trigonometri ft rtios (nd other identities) to show tht eh eqution is n identit. 0. See mrgin.. tn X sin X 9. sin X os X tn X 0. os X sin X os X tn X Prtie - lger Find the geometri men of eh pir of numers.. nd. nd 6. nd. nd. 0 nd nd 0 lger efer to the figure to omplete eh proportion h h 0.. h. lger Find the vlues of the vriles. h h Similrit in ight ringles el-world onnetion orn tht fills the in in Eerise would mke,00 gllons of ethnol.. Error nlsis student sttes tht sin. sin X euse the lengths of Y the sides of # re greter thn the lengths of the sides of #XYZ. Is the student orret Eplin. Z X No; the > re M nd the sine rtio for is onstnt esson - Sine nd osine tios. he ltitude to the hpotenuse of right tringle divides the hpotenuse into segments 6 in. nd 0 in. long. Find the length h of the ltitude.. sin X os X hp. dj. hp. dj. tn X 9. os X tn X dj. hp. dj. hp. sin X 0. sin X tn X hp. dj. dj. hp. os X

4 . ssess & eteh esson Qui Use this figure for Eerises nd.. Write the rtios for sin nd 6 0 sin. sin, sin. Write the rtios for os nd 0 6 os. os, os Use this figure for Eerises nd.. Find to the nerest tenth..0. Find to the nerest tenth..6 Use this figure for Eerises nd P N Find to the nerest degree. 6. Find to the nerest degree. 6 lterntive ssessment Hve students write two mesurement prolems involving distnes in our shool. Students lso should show how to solve one prolem using the sine rtio nd the other prolem using the osine rtio. M nline Homework Help Visit: PHShool.om We ode: ue-00. he re equl; es; he sine nd osine of omplementr ' re.. Smple: osine of l sine of the ompl. of l.. Yes; use n trig. funtion nd the known mesures to find one other side. Use the Pthgoren hm. to find the rd side. Sutrt the ute l mesure from 90 to get the other l mesure. Find the vlues of w nd then. ound lengths to the nerest tenth nd ngle mesures to the nerest degree.... PS w 0 w 6 0 w w ;. w ;. w 6.;.6.. In #, how does sin ompre to os Is this true for the ute ngles of other right tringles. eding Mth he word osine is derived See left. 6 from the words omplement s sine (see pge 69). Whih ngle in # is the omplement of & 0 Of & l; l. Eplin wh the derivtion of the word osine mkes sense. See left. 6. Find eh rtio. P. sin P ". os P.. " ". d. ". sin d. os e. Mke onjeture out how the sine nd osine of ngle re relted. he re equl.. Writing eon sid tht if she hd digrm tht showed the mesure of one ute ngle nd the Q length of one side of right tringle, she ould find the mesure of the other ute ngle nd the lengths of the other sides. Is she orret Eplin. See left.. Find eh rtio.. sin S. os S. sin.. " " d. os e. Mke onjeture out how the sine nd e. os 0 " sin 0 osine of 0 ngle re relted. See left. f. sin 60 " os 60 f. Mke onjeture out how the sine nd 60 osine of 60 ngle re relted. See left. S Proof 9. For right # with right &, prove eh of the following.. sin, no mtter how lrge & is.. See mrgin.. os, no mtter how smll & is. 0 d. nswers m vr. Smples re given. 0. sin X for X 9.9; no PHShool.om For: rphing lultor proedures We ode: ue- hllenge hpter ight ringles nd rigonometr 0. rphing lultor Use the E feture of our grphing lultor to stud sin X s X gets lose (ut ) to 90. In the Y sreen, enter Y sin X.. Use the SE feture so tht X strts t 0 nd hnges. ess the E. From the tle, wht is sin X for X Perform numeril oom in. Use the SE feture, so tht X strts with 9 nd hnges 0.. Wht is sin X for X 9.9. ontinue to numerill oom in on vlues lose to 90. Wht is the gretest vlue ou n get for sin X on our lultor How lose is X to 90 Does our result ontrdit wht ou re sked to prove in Eerise 9 See left. d. Writing Use right tringles to eplin the ehvior of sin X found ove. See mrgin. Show tht eh eqution is n identit showing tht eh epression on the left simplifies to.. See mrgin.. (sin ) + (os ). (sin ) + (os ). - (tn ). - ( os ) ( sin ) ( tn ). Show tht (tn ) - (sin ) (tn ) (sin ) is n identit. 0 N See mrgin. 9. nswers m vr. Smples re given.. Sine sin hp., if sin, then hp., whih is impossile. dj.. Sine os hp., if os, then dj. hp., whih is impossile. 0. d. For ' tht pproh 90, the side gets lose to the hp. in length, so hp. pprohes.

5 el-world onnetion Polnd honored opernius with this 000-lot note, lst used in 99. est Prep 6. stronom opernius devised method different from the one in Emple in order to find the sies of the orits of plnets frther from the sun thn Erth. His method involved noting the numer of ds etween the times tht plnet ws in the positions leled nd in the digrm. Using this time nd the numer of ds in eh plnet s er, he lulted nd d.. For Mrs,. nd d 0.. How fr is Mrs from the sun in stronomil units (U) out. U. For Jupiter,.9 nd d 00.. How fr is Jupiter from the sun in stronomil units Outer plnet s orit Erth s orit d Sun not to sle U out. U est Prep esoures For dditionl prtie with vriet of test item formts: Stndrdied est Prep, p. 6 est-king Strtegies, p. 60 est-king Strtegies with rnsprenies Eerise 6 Point out tht opernius s method depends on the sun, Erth, nd outer plnets eing in line t one point in time nd forming right ngle t the other point in time. Multiple hoie Short esponse Mied eview. Wht is the vlue of to the nerest whole numer... D. 6. Wht is the vlue of to the nerest tenth F.... H..6 J.. 9. Wht is the vlue of to the nerest whole numer... D. 0. Use the figure t the right.. Find m&. Show our work.. See mrgin.. Find m& two different methods. Show our work. H (sin ) ± (os ) Q ± ± Q. (os ) (tn ) ( ). (sin ) (tn ) Q Q for Help esson - esson - esson 6- lesson qui, PHShool.om, We ode: u-00. (sin ) ± (os ) Q ± ± Q Find the vlue of. ound nswers to the nerest tenth he wll of room is in the shpe of golden retngle. If the height of the wll is ft, wht re the possile lengths of the wll to the nerest tenth.9 ft or.9 ft Find the vlue of for eh prllelogrm esson - Sine nd osine tios. (tn ) (sin ) ( ) () () (tn ) (sin ) () () 0. []. os 0 ml os ( 0) N 6. ml N 90 6 O ml sin ( 0) N [] one ngle found orretl

### 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

### EXAMPLE EXAMPLE. Quick Check EXAMPLE EXAMPLE. Quick Check. EXAMPLE Real-World Connection EXAMPLE

- Wht You ll Lern To use the Pthgoren Theorem To use the onverse of the Pthgoren Theorem... nd Wh To find the distne etween two doks on lke, s in Emple The Pthgoren Theorem nd Its onverse hek Skills You

### 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

### 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

### 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

### 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 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

### 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

### 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

### 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

### 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

### 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 2.1 Inductive Reasoning

Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10,

### 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 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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?

### 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

### 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

### 84 cm 30 cm. 12 in. 7 in. Proof. Proof of Theorem 7-4. Given: #QXY with 6 Prove: * RS * XY

-. Pln Ojetives o use the ie-plitter heorem o use the ringle-ngle- isetor heorem Emples Using the ie-plitter heorem el-worl onnetion Using the ringle-ngle- isetor heorem Mth kgroun - Wht ou ll Lern o use

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Section 5-5 Solving Right Triangles*

5-5 Solving Right Tringles 379 79. 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

### 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

### 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

### 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

### 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.

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Pre-algebra 7* In your group consider the following problems:

Pre-lger * Group Activit # Group Memers: In our group consider the following prolems: 1) If ever person in the room, including the techer, were to shke hnds with ever other person ectl one time, how mn

### 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

### Unit 6: Exponents and Radicals

Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -

### 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

### 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

### 1.1 THE CARTESIAN PLANE AND THE DISTANCE FORMULA

6000_00.qd //05 : PM Pge CHAPTER Funtions, Grphs, nd Limits. THE CARTESIAN PLANE AND THE DISTANCE FORMULA Plot points in oordinte plne nd red dt presented grphill. Find the distne etween two points in

### 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

### 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

### 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

### Quick Guide to Lisp Implementation

isp Implementtion Hndout Pge 1 o 10 Quik Guide to isp Implementtion Representtion o si dt strutures isp dt strutures re lled S-epressions. The representtion o n S-epression n e roken into two piees, the

### 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

### 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

### Chapter. Radicals (Surds) Contents: A Radicals on a number line. B Operations with radicals C Expansions with radicals D Division by radicals

Chter 4 Rdils (Surds) Contents: A Rdils on numer line B Oertions with rdils C Exnsions with rdils D Division y rdils 88 RADICALS (SURDS) (Chter 4) INTRODUCTION In revious yers we used the Theorem of Pythgors

### 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

### The Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center

Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,

### GRADE 4. Fractions WORKSHEETS

GRADE Frtions WORKSHEETS Types of frtions equivlent frtions This frtion wll shows frtions tht re equivlent. Equivlent frtions re frtions tht re the sme mount. How mny equivlent frtions n you fin? Lel eh

### 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

### 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

### Chess and Mathematics

Chess nd Mthemtis in UK Seondry Shools Dr Neill Cooper Hed of Further Mthemtis t Wilson s Shool Mnger of Shool Chess for the English Chess Federtion Mths in UK Shools KS (up to 7 yers) Numers: 5 + 7; x

### 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

### 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

### Review. Scan Conversion. Rasterizing Polygons. Rasterizing Polygons. Triangularization. Convex Shapes. Utah School of Computing Spring 2013

Uth Shool of Computing Spring 2013 Review Leture Set 4 Sn Conversion CS5600 Computer Grphis Spring 2013 Line rsteriztion Bsi Inrementl Algorithm Digitl Differentil Anlzer Rther thn solve line eqution t

### 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

### 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

### 4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A

Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter

### 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

### 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

### 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

### 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

### This unit will help you to calculate perimeters and areas of circles and sectors, and to find the radius given the circumference or area.

Get strte 1 Cirles This unit will help you to lulte perimeters n res of irles n setors, n to fin the rius given the irumferene or re. AO1 Flueny hek 1 Roun 4.635 to 2 eiml ples (.p.) 2 Roun 5.849 to 1.p.

### a 2 + b 2 = c 2. There are many proofs of this theorem. An elegant one only requires that we know that the area of a square of side L is L 2

Pythgors Pythgors A right tringle, suh s shown in the figure elow, hs one 90 ngle. The long side of length is the hypotenuse. The short leg (or thetus) hs length, nd the long leg hs length. The theorem

### Special Right Triangles. Use a protractor to find the measures of the angles of each triangle

- Wht You ll Lern To ue the propertie of 5-5-90 tringle To ue the propertie of 0-60-90 tringle... nd Wh To find the ditnce from home plte to econd e on oftll dimond, in Emple Specil Right Tringle Check

### Heron, Brahmagupta, Pythagoras, and the Law of Cosines

University of Nersk - Linoln DigitlCommons@University of Nersk - Linoln MAT Exm Expository Ppers Mth in the Middle Institute Prtnership 7-1-006 Heron, Brhmgupt, Pythgors, nd the Lw of Cosines Kristin K.

### Lesson 4.1 Triangle Sum Conjecture

Lesson 4.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q 2., y 3., b 31 82 p 98 q 28 53 y 17 79 23 50 b 4. r, s, 5., y 6. y t t s r 100 85 100 y 30 4 7 y 31 7. s 8.

### 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

### Muscle and Joint Forces II

Musle n Joint Fores II Leture outline! mehnis of the hip uring single legge stne (ontinue)! efinition of musle moments! effet of joint ngle on moment rm! sttill ineterminte sstems Stephen Roinovith, Ph.D.

### 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

### 1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator

AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.

### Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you

### 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

### x 2 New Vocabulary congruent polygons EXAMPLE #TJD > #RCF. List the congruent corresponding parts. Angles: &T > &R &J > &C &D > &F Quick Check

-. ln Objectives o recognize congruent figures nd their corresponding prts xmples ming ongruent rts 2 el-world onnection inding ongruent ringles roving ringles ongruent - Wht ou ll ern o recognize congruent

### Proving the Pythagorean Theorem

CONCEPT DEVELOPMENT Mthemtis Assessment Projet CLASSROOM CHALLENGES A Formtive Assessment Lesson Proving the Pythgoren Theorem Mthemtis Assessment Resoure Servie University of Nottinghm & UC Berkeley For

### 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

### P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn

33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of

### 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