Non Right Angled Triangles

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1 Non Right ngled Tringles Non Right ngled Tringles urriulum Redy

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3 Non Right ngled Tringles NON RIGHT NGLED TRINGLES sin i, os i nd tn i re lso useful in non-right ngled tringles. This unit will show you how to use these trigonometri rtios to find the sizes of ngles nd length of sides. Try to nswer these questions now, efore working through the hpter. I used to think: How re sides leled in reltion to the leling ngles of non right ngled tringle? Drw nd lel tringle. The "osine rule" is used when two sides of the tringle nd the ngle etween them is given Wht is this rule? nswer these questions fter you hve worked through the hpter. ut now I think: How re sides leled in reltion to the leling ngles of non right ngled tringle? Drw nd lel tringle. The "osine rule" is used when two sides of the tringle nd the ngle etween them is given Wht is this rule? Wht do I know now tht I didn t know efore? 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 TOPI

4 Non Right ngled Tringles sis Leling Tringles Everyone lels tringles the sme wy. Eh ngle is leled with pitl letter, nd their opposite sides re leled with the sme letter, ut non-pitl. For T Opposite to Opposite to Opposite to For TLMN Opposite to m Opposite to n M l N n m L Opposite to l K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

5 Non Right ngled Tringles Questions sis. Lel the sides in the following tringles. X Y J H K W. Lel the ngles in the following the tringles. m q p g f r 3. omplete the leling of the following tringles. T u M k s L 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 3 TOPI

6 Non Right ngled Tringles Knowing More So fr we've used trigonometry in right-ngled tringles, ut we n tully use it to find sides nd ngles in ny tringle! The re Rule T is not right-ngled. The perpendiulr height, h, is drwn in. ` re # se # height re # # h We nsee: h sin ` h sin re # # sin h ` re sin So the re is hlf the produt of two sides nd the sine of their interior ngle. ny two sides nd their interior ngle will work: re sin or re sin or re sin Find the re of T DEF to deiml ple D 0 m 00 E F 4 m d nd f re given, so we will use re df sin E re TDEF dfsin E ^ 4 h^ 0 hsin f. 9. 7m ^deimlpleh Find the re of T to deiml ple nd re given, so we will use re re T sin sin (write this in T ) ^ h^ 8 hsin m ( deiml ple) 4 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

7 Non Right ngled Tringles Questions Knowing More. Find the re of the following tringles to deiml ple. 7 m m 0 m 6.4 m 78. T hs , ,. 4mnd. m. Drw the tringle. Find + to nerest degree. Find the re of the tringle to deiml ple. 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 5 TOPI

8 Non Right ngled Tringles Questions Knowing More 3. n you find the res of the following shpes to the nerest squre m? E 0 m 3 6 m D 3 m Hint: Drw in Digonl D m 9.5 m m 0 m 93 6 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

9 Non Right ngled Tringles Questions Knowing More 4. Luke wnts to mke kite in the shpe elow. How mny squre m of mteril will he need? m m 44 D 5. The re of the tringle T PQR is 38.6 m. Find + Q. (Hint: Find+ P first) Q 0 m P m R 00% Non Right ngled Tringles K 8 7 Mthletis 00% 3P Lerning TOPI

10 Non Right ngled Tringles Using Our Knowledge The Sine Rule In ny tringle T we hve the formuls: sin sin sin or sin sin sin The proof is different depending on whether the tringle is ute or otuse. If T is n ute tringle If T is n otuse tringle h h D 80 - D In TD: h sin ` h sin In TD: h sin ` h sin In TD: ` sin sin h sin ` h sin (oth equl h) In TD: h sin^80 -h ` h sin^80 -h ` h sin Rememer sin^80 - θh sin θ ` sin sin ` sin sin (oth equl h) If we drew the ltitude from to then we ould show ` sin sin sin sin If we drew the ltitude from to then we ould show sin sin 8 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

11 Non Right ngled Tringles Using Our Knowledge Find the vlue of in T to deiml ples 0 m 8 7 m 8 Using the Sine Rule: ` sin sin 0 sin8 sin 8 ` 0 sin 8 sin 8 0^0469. fh f m f ( deiml ples) Find the vlue of + E in T DEF to the nerest degree D 36 E Using the Sine Rule: m F 3 m sine sin D e d ` sine sin 36 3 ` sin E 3 sin f - ` E sin ^0. 694fh f. 44 (nerest degree) Use the digrm elow to nswer these questions L Find + LPN to the nerest degree. N 55 0 m 3 9 m P 7 m M sin+ LPN sin + LNP LN LP ` sin+ LPN sin ` sin + LPN 0 sin ` + LPN sin ( ) ( nerest degree) Find the length of LM to the nerest m. First, find + LPM + LPM LPN Now, use the sine rule LM PM sin+ LPM sin + PLM ` LM 7 sin4 sin 3 ` LM 7sin m (nerest m) sin 3 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 9 TOPI

12 Non Right ngled Tringles Questions Using Our Knowledge The Sine Rule. Find the length y in the tringles elow to deiml ple. m y 6 37 y 9 m 4 0. Find the size of ngle in the following tringles: 9 m m 0 6 m 46 m 0 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

13 Non Right ngled Tringles Questions Using Our Knowledge 3. ship hs sils s drwn in the digrm elow: D 5 0 m 85 Find the length of. Find the lengths of D nd D. 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 TOPI

14 Non Right ngled Tringles Using Our Knowledge The osine Rule In ny tringle T we hve the formuls + - os or + - os or + - os The proof is different depending on if the tringle is ute or otuse. If T is n ute tringle If T is n otuse tringle h h x D (-x) D (x- ) x In T D: h + ^- xh h + x + -x (Pythgors) In T D: h + ^x- h h + x + -x (Pythgors) Susitute Susitute Susitute Susitute In T D: h + x (Pythgors) In T D: h + x (Pythgors) nd os x ` x os nd os x ` x os + - os + - os If the h is drwn from other orners we n show + - os or + - os If the h is drwn from other orners we n show + - os or + - os K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

15 Non Right ngled Tringles Using Our Knowledge Find the vlue of in T to deiml ples 8 m 30 5 m Using the osine Rule: ` ` + - os ^ h^8hos ^0. 866fh ` 9. 77f ` m ( deiml ples) Find the vlue of + Q to the nerest degree P 6 m 5 m Using the osine Rule: q p + r - pros Q ` ^0h^5hos Q R 0 m Q Mke os Q the sujet 0 ^ h^5hos Q os Q ` os Q f 300 Solve for + Q - + Q os ^ h 5. 56f. 6 (nerest degree) Find + in the following to deiml ple D 49. D + D - # D # D # os D ^ h^hos 49. This is the osine rule 5 m m f 9. 49fm Write this in the digrm + - # # # os 6 m 8 m ^9. 49fh ^6h^8 hos ^9. 49fh ` os 6 ^ h^8h f ` + os - ^0347. fh f ( deiml ple) 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 3 TOPI

16 Non Right ngled Tringles Questions Using Our Knowledge The osine Rule 4. Find the length of the missing side x in eh of the following to deiml ple. D P 9 m x 5 m 73 x E m F Q 5 m 56 R 5. Find the vlue of the ngle in eh of the following to the nerest degree. 50 m 47 m 55 m 39 m W 6 m Y X 3 m 4 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

17 Non Right ngled Tringles Questions Using Our Knowledge 6. nyon hs the following dimensions: N 46 m L m M 50 m P To wlk round this nyon, would it e shorter to wlk long LNM or LPM? Use working to support your nswer. 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 5 TOPI

18 Non Right ngled Tringles Questions Using Our Knowledge oth Rules 7. Use the sine nd osine rules to find the missing sides nd ngles in these tringles to deiml ples m 67 Q 38 m 55 m P 4 R 6 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

19 Non Right ngled Tringles Thinking More n miguous se For The Sine Rule Rememer tht sini sin(80-i). So when the sine rule is used to find n ngle (using sin - ) two nswers re possile. For exmple if sini 0.5 then i 30 or 50 (80-30). It is our jo to determine whether one or oth nswers re vlid. This depends on whether third ngle exists or not. Find + (to deiml ple) in T if + 50, 4m nd 7. m ording to the sine rule sin sin ` sin sin ` sin 7. # sin ` + sin ^094. h - ` or dd eh nswer to the given ngle. If this sum is less thn 80 then the nswer is vlid euse the third ngle exists. The given ngle in the question is + 50 ` ` vlid or ` vlid oth sums re less thn 80, nd so oth nswers for + re vlid. So T my look like either of these tringles: 4 m m 50 or 4 m m 50 se where oth nswers re vlid (like ove) is sid to e "miguous." 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 7 TOPI

20 Non Right ngled Tringles Thinking More Determining if nswers re Vlid Of the two possile nswers, one will lwys e otuse So if you know the tringle is ute, then the otuse ngle is invlid. If you know the missing ngle is otuse, then the ute ngle is invlid. (90to80 ) nd the other will lwys e ute ^0to 90h. The sum of ngles in tringle is 80. So if the sum of possile nswer nd the given ngle is 80 or more it mens tht the third ngle in the tringle is 0 or negtive. This is impossile, so the nswer would e invlid. Find + L (to deiml ple) in T LMN if + N 80, n m nd l m ording to the sine rule sinl sin N l n ` sinl sin ` sin L 40. 5# sin f ` + L sin ^0806. fh ` + L 55. 5f. 55. or +L dd eh nswer to the given ngle ^+ N 80h. ` + L+ + N ` vlid or + L+ + N ` invlid So only the first nswer is vlid nd + L 55.. This is not n miguous se. + L 55. nd T LMN looks like this M 40.5m 48.6m N L 8 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

21 Non Right ngled Tringles Questions Thinking More. In tringle T 9m, 5m nd + 0. Find two possile vlues for + to the nerest degree. hek if these nswers re vlid. Is this n miguous se? Drw rough sketh of the tringle(s). d Find the third ngle in the tringle(s) nd dd them to the digrm in. 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 9 TOPI

22 Non Right ngled Tringles Questions Thinking More. In T PQR, + R 40, p 0.3m nd r Find the two possile vlues for + P to the nerest degree. Is this n miguous se (re oth ngles vlid)? Whih nswer is vlid if T PQR is ute? Whih nswer is invlid? d Whih nswer is vlid if + P is otuse? Whih nswer is invlid? 0 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

23 Non Right ngled Tringles Questions Thinking More 3. In tringle T JKL, + K 70, k 5. 4m nd l 4m. Find the vlid size(s) of + L. Find + J. Find the length of j. d Drw T JKL. 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 TOPI

24 Non Right ngled Tringles nswers sis: sis:. 3. X w Y T u y x s S t W H k j K h J U l M m k L. K Knowing More:. re T 8. 5m R re T 74. m q p. P F m r Q G m.m g f re T 89. 0m M K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

25 Non Right ngled Tringles nswers Knowing More: Thinking More: 3. Totl re m. + 6 or + 54 Totl re 3m oth nswers re vlid; therefore this is n miguous se. 4. Totl re 99m Q m 5m Using Our Knowledge: 0 6. y 9. m y. 7m 5m m 0 40 d + 34 or m(to d.p.) or + 0 D. 0.0m nd D. 6.0m oth nswers re vlid; therefore this is n miguous se. 4. x 9.6m (to d.p.) x.5m (tod.p.) If PQR is ute, the ngles re in the rnge 0-90, +P 70 is the vlid nswer nd + P 0 is invlid (tonerest degree) (tonerest degree) d If PQR is otuse, the ngles re in the rnge 90-80, +P 0 is the vlid nswer nd +P 70 is invlid. 6. It would e shorter to wlk long LPM thn LNM 3. + L 50 +J m (tod.p.) 9.89m (tod.p.) q 7.0m (tod.p.) d j 47. 3m K 70 4m 47.3m + Q (tod.p.) + R (tod.p.) J m 50 L 00% Non Right ngled Tringles Mthletis 00% 3P Lerning K 8 3 TOPI

26 Non Right ngled Tringles Notes 4 K 8 00% Non Right ngled Tringles TOPI Mthletis 00% 3P Lerning

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Part I: Study the theorem statement. Nme 1 Nme 2 Nme 3 A STUDY OF PYTHAGORAS THEOREM Instrutions: Together in groups of 2 or 3, fill out the following worksheet. You my lift nswers from the reding, or nswer on your own. Turn in one pket for

12.4 Similarity in Right Triangles Nme lss Dte 12.4 Similrit in Right Tringles Essentil Question: How does the ltitude to the hpotenuse of right tringle help ou use similr right tringles to solve prolems? Eplore Identifing Similrit in Right

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

Precalculus Notes: Unit 6 Law of Sines & Cosines, Vectors, & Complex Numbers. A can be rewritten as Dte: 6.1 Lw of Sines Syllus Ojetie: 3.5 Te student will sole pplition prolems inoling tringles (Lw of Sines). Deriing te Lw of Sines: Consider te two tringles. C C In te ute tringle, sin In te otuse tringle,

m m m m m m m m P m P m ( ) m m P( ) ( ). The o-ordinte of the point P( ) dividing the line segment joining the two points ( ) nd ( ) eternll in the r CO-ORDINTE GEOMETR II I Qudrnt Qudrnt (-.+) (++) X X - - - 0 - III IV Qudrnt - Qudrnt (--) - (+-) Region CRTESIN CO-ORDINTE SSTEM : Retngulr Co-ordinte Sstem : Let X' OX nd 'O e two mutull perpendiulr

On the diagram below the displacement is represented by the directed line segment OA. Vectors Sclrs nd Vectors A vector is quntity tht hs mgnitude nd direction. One exmple of vector is velocity. The velocity of n oject is determined y the mgnitude(speed) nd direction of trvel. Other exmples

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

Area and Perimeter. Area and Perimeter. Curriculum Ready. Are nd Perimeter Curriculum Redy www.mthletics.com This ooklet shows how to clculte the re nd perimeter of common plne shpes. Footll fields use rectngles, circles, qudrnts nd minor segments with specific

Perimeter and Area. Mathletics Instant Workbooks. Copyright Perimeter nd Are Student Book - Series J- L B Mthletis Instnt Workooks Copyright Student Book - Series J Contents Topis Topi - Plne shpes Topi 2 - Perimeter of regulr shpes Topi 3 - Perimeter of irregulr

Comparing the Pre-image and Image of a Dilation hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity

9.1 Day 1 Warm Up. Solve the equation = x x 2 = March 1, 2017 Geometry 9.1 The Pythagorean Theorem 1 9.1 Dy 1 Wrm Up Solve the eqution. 1. 4 2 + 3 2 = x 2 2. 13 2 + x 2 = 25 2 3. 3 2 2 + x 2 = 5 2 2 4. 5 2 + x 2 = 12 2 Mrh 1, 2017 Geometry 9.1 The Pythgoren Theorem 1 9.1 Dy 2 Wrm Up Use the Pythgoren

Proportions: A ratio is the quotient of two numbers. For example, 2 3 Proportions: rtio is the quotient of two numers. For exmple, 2 3 is rtio of 2 n 3. n equlity of two rtios is proportion. For exmple, 3 7 = 15 is proportion. 45 If two sets of numers (none of whih is 0)

10. AREAS BETWEEN CURVES . AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

R(3, 8) P( 3, 0) Q( 2, 2) S(5, 3) Q(2, 32) P(0, 8) Higher Mathematics Objective Test Practice Book. 1 The diagram shows a sketch of part of Higher Mthemtics Ojective Test Prctice ook The digrm shows sketch of prt of the grph of f ( ). The digrm shows sketch of the cuic f ( ). R(, 8) f ( ) f ( ) P(, ) Q(, ) S(, ) Wht re the domin nd rnge of

S2 (2.2) Pythagoras.notebook March 04, 2016 S2 (2.2) Pythgors.noteook Mrh 04, 2016 Dily Prtie 16.12.2015 Q1. Multiply out nd simplify 9x 3(2x + 1) Q2. Solve the eqution 3(2x + 4) = 18 Q3. If 1 = \$1.30, how muh is 50 in dollrs? Tody we will e lerning

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. Mth 3329-Uniform Geometries Leture 06 1. Review of trigonometry While we re looking t Eulid s Elements, I d like to look t some si trigonometry. Figure 1. The Pythgoren theorem sttes tht if = 90, then

MCH T 111 Handout Triangle Review Page 1 of 3 Hnout Tringle Review Pge of 3 In the stuy of sttis, it is importnt tht you e le to solve lgeri equtions n tringle prolems using trigonometry. The following is review of trigonometry sis. Right Tringle:

Algebra & Functions (Maths ) opposite side Instructor: Dr. R.A.G. Seel Trigonometr Algebr & Functions (Mths 0 0) 0th Prctice Assignment hpotenuse hpotenuse side opposite side sin = opposite hpotenuse tn = opposite. Find sin, cos nd tn in 9 sin

Pythagoras Theorem PYTHAGORAS THEOREM. Pthgors Theorem PYTHAGORAS THEOREM www.mthletis.om.u How oes it work? Solutions Pthgors Theorem Pge 3 questions Right-ngle tringles D E x z Hotenuse is sie: F Hotenuse is sie: DF Q k j l Hotenuse is sie:

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved

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

Basic Angle Rules 5. A Short Hand Geometric Reasons. B Two Reasons. 1 Write in full the meaning of these short hand geometric reasons. si ngle Rules 5 6 Short Hnd Geometri Resons 1 Write in full the mening of these short hnd geometri resons. Short Hnd Reson Full Mening ) se s isos Δ re =. ) orr s // lines re =. ) sum s t pt = 360. d)

Special Numbers, Factors and Multiples Specil s, nd Student Book - Series H- + 3 + 5 = 9 = 3 Mthletics Instnt Workooks Copyright Student Book - Series H Contents Topics Topic - Odd, even, prime nd composite numers Topic - Divisiility tests

K 7. Quadratic Equations. 1. Rewrite these polynomials in the form ax 2 + bx + c = 0. Identify the values of a, b and c: Qudrti Equtions The Null Ftor Lw Let's sy there re two numers nd. If # = then = or = (or oth re ) This mens tht if the produt of two epressions is zero, then t lest one of the epressions must e equl to

Probability. b a b. a b 32. Proility If n event n hppen in '' wys nd fil in '' wys, nd eh of these wys is eqully likely, then proility or the hne, or its hppening is, nd tht of its filing is eg, If in lottery there re prizes nd lnks,

9.5 Start Thinking. 9.5 Warm Up. 9.5 Cumulative Review Warm Up 9.5 Strt Thinking In Lesson 9.4, we discussed the tngent rtio which involves the two legs of right tringle. In this lesson, we will discuss the sine nd cosine rtios, which re trigonometric rtios for cute

青藜苑教育 The digrm shows the position of ferry siling between Folkestone nd lis. The ferry is t X. X 4km The pos 青藜苑教育 www.thetopedu.com 010-6895997 1301951457 Revision Topic 9: Pythgors Theorem Pythgors Theorem Pythgors Theorem llows you to work out the length of sides in right-ngled tringle. c The side opposite

MTH 505: Number Theory Spring 2017 MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of \$ nd \$ s two denomintions of coins nd \$c

Polynomials and Division Theory Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the QUADRATIC EQUATION Contents Topi Pge No. Theory 0-04 Exerise - 05-09 Exerise - 09-3 Exerise - 3 4-5 Exerise - 4 6 Answer Key 7-8 Syllus Qudrti equtions with rel oeffiients, reltions etween roots nd oeffiients,

I1.1 Pythagoras' Theorem. I1.2 Further Work With Pythagoras' Theorem. I1.3 Sine, Cosine and Tangent. I1.4 Finding Lengths in Right Angled Triangles UNIT I1 Pythgors' Theorem nd Trigonometric Rtios: Tet STRAND I: Geometry nd Trigonometry I1 Pythgors' Theorem nd Trigonometric Rtios Tet Contents Section I1.1 Pythgors' Theorem I1. Further Work With Pythgors'

= x x 2 = 25 2 9.1 Wrm Up Solve the eqution. 1. 4 2 + 3 2 = x 2 2. 13 2 + x 2 = 25 2 3. 3 2 2 + x 2 = 5 2 2 4. 5 2 + x 2 = 12 2 Mrh 7, 2016 Geometry 9.1 The Pythgoren Theorem 1 Geometry 9.1 The Pythgoren Theorem 9.1

Section 6: Area, Volume, and Average Value Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

For a, b, c, d positive if a b and. ac bd. Reciprocal relations for a and b positive. If a > b then a ab > b. then Slrs-7.2-ADV-.7 Improper Definite Integrls 27.. D.dox Pge of Improper Definite Integrls Before we strt the min topi we present relevnt lger nd it review. See Appendix J for more lger review. Inequlities: QUADRATIC EQUATIONS OBJECTIVE PROBLEMS +. The solution of the eqution will e (), () 0,, 5, 5. The roots of the given eqution ( p q) ( q r) ( r p) 0 + + re p q r p (), r p p q, q r p q (), (d), q r p q.

Geometry. Trigonometry of Right Triangles. Slide 2 / 240. Slide 1 / 240. Slide 3 / 240. Slide 4 / 240. Slide 6 / 240. Slide 1 / 240 Slide 2 / 240 New Jerse enter for Tehing nd Lerning Progressive Mthemtis Inititive This mteril is mde freel ville t www.njtl.org nd is intended for the non-ommeril use of students nd tehers.

Applications of trigonometry 3 3 3 3 3D 3E 3F 3G 3H Review of right-ngled tringles erings Using the sine rule to find side lengths Using the sine rule to find ngles re of tringle Using the osine rule to find side lengths Using the

April 8, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then. pril 8, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm shown in the Figure Sine its digonls hve equl length,, the prllelogrm

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

GEOMETRICAL PROPERTIES OF ANGLES AND CIRCLES, ANGLES PROPERTIES OF TRIANGLES, QUADRILATERALS AND POLYGONS: GEOMETRICL PROPERTIES OF NGLES ND CIRCLES, NGLES PROPERTIES OF TRINGLES, QUDRILTERLS ND POLYGONS: 1.1 TYPES OF NGLES: CUTE NGLE RIGHT NGLE OTUSE NGLE STRIGHT NGLE REFLEX NGLE 40 0 4 0 90 0 156 0 180 0

2. There are an infinite number of possible triangles, all similar, with three given angles whose sum is 180. SECTION 8-1 11 CHAPTER 8 Setion 8 1. There re n infinite numer of possile tringles, ll similr, with three given ngles whose sum is 180. 4. If two ngles α nd β of tringle re known, the third ngle n e found

March 26, 2017 Math 9. Geometry. Solving vector problems. Problem. Prove that if vectors and satisfy, then. Mrh 26, 2017 Mth 9 Geometry Solving vetor prolems Prolem Prove tht if vetors nd stisfy, then Solution 1 onsider the vetor ddition prllelogrm D shown in the Figure Sine its digonls hve equl length,, the

Sect 10.2 Trigonometric Ratios 86 Sect 0. Trigonometric Rtios Objective : Understnding djcent, Hypotenuse, nd Opposite sides of n cute ngle in right tringle. In right tringle, the otenuse is lwys the longest side; it is the side opposite

5Trigonometric UNCORRECTED PAGE PROOFS. ratios and their applications 5Trigonometri rtios nd their pplitions 5.1 Kik off with CS 5.2 Trigonometry of right-ngled tringles 5.3 Elevtion, depression nd erings 5.4 The sine rule 5.5 The osine rule 5.6 rs, setors nd segments 5.7

Vectors. a Write down the vector AB as a column vector ( x y ). A (3, 2) x point C such that BC = 3. . Go to a OA = a Streth lesson: Vetors Streth ojetives efore you strt this hpter, mrk how onfident you feel out eh of the sttements elow: I n lulte using olumn vetors nd represent the sum nd differene of two vetors grphilly. QUADRATIC EQUATION EXERCISE - 0 CHECK YOUR GRASP. Sine sum of oeffiients 0. Hint : It's one root is nd other root is 8 nd 5 5. tn other root 9. q 4p 0 q p q p, q 4 p,,, 4 Hene 7 vlues of (p, q) 7 equtions

5. Every rational number have either terminating or repeating (recurring) decimal representation. CHAPTER NUMBER SYSTEMS Points to Rememer :. Numer used for ounting,,,,... re known s Nturl numers.. All nturl numers together with zero i.e. 0,,,,,... re known s whole numers.. All nturl numers, zero nd

A B= ( ) because from A to B is 3 right, 2 down. 8. Vectors nd vector nottion Questions re trgeted t the grdes indicted Remember: mgnitude mens size. The vector ( ) mens move left nd up. On Resource sheet 8. drw ccurtely nd lbel the following vectors.

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let Lesson 2.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 12, 16,, 2. 400, 200, 100, 50, 25,, 3. 1 8, 2 7, 1 2, 4, 5, 4. 5, 3, 2, Curriulum Redy ACMNA: 33 75 76 77 79 www.mthletis.om Fill in the spes with nything you lredy know out Alger Creer Opportunities: Arhitets eletriins plumers et. use it to do importnt lultions. Give this