DOCSLIB.ORG

Algebra and Trigonometry Review Material

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algebra and Trigonometry Review Material Algebra and Trigonometry Review Material Department of Mathematics Vanderbilt University June 2, 2004 http://atlas.math.vanderbilt.edu/~pscrooke/calculus/calculus.html ¡ ¢ £ ¤ ¢ £ ¥ ¦ § ¨ © ¨ ¨ ¦ © ! " ¦ # $ " ! % # & ' ¨ % ( ! ¨ ) $ % * © " + , ' ¨ # - " © #( . / 0 § ¨ © %# 1 ¤ 6 ¥ 7 ¢ 8 9 ¥ ¡ : £ ¤ ; 7 : ¤ ¥ 2 3 4 5 5 3 3 < = > ? @AB CDEFGHI JKLM L NGHM EJOHPLQP HOKG Q E LPRGE LP IT I Q G PRGM LHG LE OQU PRG EOIP 1 1 1 S 1 S FLI LQV DIGWDK E LPRGE LP LK OFXG PIY ZG LKK PRLP L OKKG P OQ OW OFXG PI I LKKGV L 1 S 1 S S S S S 1 S 1 S Y [ IGP LQ FG VGI H FGV FM K IP QU PI GKGEGQPIT LI Q \ ] ^ _ ` a ` b ` c d T OH FM U N QU L S S 1 1 1 1 1 1 1 JHOJGHPM PRLP OEJKGPGKM VGPGHE QGI PRG GKGEGQPIT ID R LI \ ] ^ e f e I LQ GNGQ QDEFGH S 1 S 1 F G P g G G Q h LQV i d Y j R I g L M OW g H P QU L IGP I LKKGV Y j R G IGPI OW QDEFGHI 1 1 1 1 S g G g KK FG DI QU Q LK DKDI LHG PRG W O K K O g Q U k 1 1 1 S S 1 j R G IGP l OW I PRG IGP OW ODQP QU QDEFGHI OH JOI P NG g R O K G Q D E F G H I k 1 S 1 1 1 ¨ ¨ © l ] ^ h ` _ ` m ` n n d I PRG IGP OW LKK JOI P NG LQV QGULP NG g R O K G QDEFGHI POUGPRGH g PR j R G IGP o OW 1 1 1 1 1 " G H O k p o ] ^ n n n ` q m ` q _ ` q h ` r ` h ` _ ` m ` n n n d I PRG IGP OW LKK t D O P GQPI OW QPGUGHI g PR QOQ GHO j R G IGP s OW 1 1 1 1 ¨ © p VGQOE Q L P O H I k 1 s ] u v ` LHG QPGUGHI g PR ] r z 1 1 w w y v w x x x Q KDVGI LKK OW PRG LFONG QDEFGHIT POUGPRGH g PR LKK PRG j R G IGP { OW 1 S 1 ¨ © x HHLP OQLK QDEFGHI K G } _ ` }~ ` T LQV Y N G H M HGLK QDEFGH e LQ FG PRODURP OW LI PRG 1 1 1 | S OH G Q P G V V IPLQ G OW L JO QP OQ L K QG WHOE PRG OH U Q Y O PRG IGP OW HGLK QDEFGHI I Q 1 1 S 1 1 1 1 1 1 V HG P OHHGIJOQVGQ G g PR L K QG OQ L JKLQGY 1 S S S 1 1 x -4 -3 -2 -10 1 2 3 4 UDHG h Y h k j R G HGLK QDEFGH K QG 1 1 G LDIG GHPL Q IDFIGPI OW PRG HGLK K QG LHG DIGV t D PG OWPGQT g G LVOJP L IJG LK S S 1 1 1 S 1 QOPLP OQ WOH PRGE PRLP I LKKGV Y j R G IGP ^ e f h e _ d OW LKK HGLK 1 1 S QDEFGHI F G P g G G Q h LQV _T Q KDV QU h LQV _T I VGQOPGV FM h ` _ Y W g G g L Q P LKK PROIG 1 S 1 1 h QDEFGHI G GJP WOH _ ` g G g H PG h ` _ ] ^ e f h e _ d Y Q UGQGHLK g G RLNG PRG W O K K O g QU S 1 1 QOPLP OQ WOH GHPL Q IGPI OW QDEFGHI LKKGV Y 1 S 1 S ` ] ^ e f e d ` ] ^ e f e d + 0 ` ] ^ e f e d + 0 ` ] ^ e f e d a b -4 -3 -2 -10 1 2 3 4 UDHG h Y _ k j R G QPGHNLK ` 1 1 j R G R O K K O g H KG LP e ] OQ PRG QDEFGH K QG LFONG I R O g I PRLP PRG QPGHNLK S 1 S 1 1 OQPL Q T g R KG PRG K K G V ¡ Q H KG LP e ] I R O g I PRLP P OQPL Q Y S 1 1 1 S 1 S 1 S 1 ¢ G LQ OQI VGH LI g G K K k S S 1 £ ` ¤ ¥ ] ^ e f e d ` ¤ ¥ ] ^ e f e d q ¥ ` ] ^ e f e d q ¥ ` ] ^ e f e d q ¥ ` ¤ ¥ ] { COPG PRLP q ¥ LQV ¤ ¥ LHG QGNGH Q KDVGV Q PRG QPGHNLKIT I Q G PRGM LHG QOP QDEFGHIY 1 S 1 1 1 S W \ LQV ¦ LHG IGPIT g G VGQOPG PRG IGP OW LKK GKGE GQPI OW \ POUGPRGH g PR LKK GKGEGQPI 1 © OW ¦ FM \ § ¦ ` g R R g G HGLV LI ¨\ DQ OQ ¦ Y 1 S 1 © ¦ ª L ^ h ` _ ` m ` a d § ^ b ` d ] ^ h ` _ ` m ` a ` b ` d F ^ h ` _ ` m ` a d § ^ h ` m ` « ` d ] ^ h ` _ ` m ` a ` « ` d COPG PRLP g G VO QOP K IP PRG ILEG GKGEGQP P g G Q IGP QOPLP OQ Y 1 1 S 1 1 _ q _ n « ` q h § r n « ` m na © ª ¬ ¬ ­ ­ ® ¯ ¬ ¬ ­ ­ ® ¯ -2.5 0.5 3.4 -4-3 -2 -1 0 1 2 3 4 UDHG h Ym k q _ n « ` h § r n « ` m na 1 q h n b ` h na § r n « ` m na ] q h n b ` m na © ª ° ° ² ± ± ³ ° ° ² ± ± ³ -1.6 0.5 1.4 3.4 -4-3-2 -1 0 1 2 3 4 UDHG h Ya k q h nb ` m na 1 < = > @ A ´ µ ¶ · ¸ ¹ º A @ B B ´ » ¼ B j R G H G LHG GHPL Q PR QUI PO GGJ Q E QV g R G Q g O H QU g PR QDEGH LK G J H G I I OQIY S 1 1 | 1 1 | 1 1 1 S 1 ¢ RGQ LVV QU OH EDKP JKM QU P g O QDEFGHI PRG OHVGH I QOP EJOHPLQPY OH G L E JKGT 1 1 1 1 1 « ¤ m ] c ] m ¤ «Y Q UGQGHLKT WOH LKK HGLK QDEFGHIT ¤ ] ¤ LQV ] © © ¨ j R G ILEG I QOP PHDG WOH IDFPHL P OQ OH V N I OQT I Q G « q m ] m q « LQV « ½ m ] m ½ « Y 1 S 1 1 1 1 1 S y y ¾ O H G O N G H T g R G Q g G LVV OH EDKP JKM QDEFGHI g G VO QOP QGGV JLHGQPRGIGI PO IJG WM PRG 1 S 1 OHVGH Q g R R PRG OJGHLP OQ IRODKV FG JGHWOHEGV T I Q G 1 1 S 1 1 S ¤ ¤ ¿ ] ¤ ¤ ¿ LQV ¿ ] ¿ D P T LUL Q T PR I I QOP PHDG WOH IDFPHL P OQ OH V N I OQ T I Q G « q m q _ ] « q h ] a T 1 1 1 S 1 1 1 1 1 S g R KG « q m q _ ] _ q _ ] r Y 1 j R D I g G QGGV JLHGQPRGIGI g R G Q g G JGHWOHE IDFPHL P OQ OH g R G Q g G VGLK g PR EOHG S 1 1 PRLQ OQG OJGHLP OQT I Q G « À m ¤ _ ] h « ¤ _ ] h T g R KG « À m ¤ _ ] « À « ] _«Y 1 1 S 1 j R I HGLPGI LQ ONGHLFDQVLQ G OW JLHGQPRGIGIT IO g G LUHGG PO JGHWOHE EDKP JK LP OQI 1 S S 1 1 S 1 m LQV V N I OQI H I P LQV PRGQ LVV P OQI LQV IDFPHL P OQIÁ EOHGONGHT g G JGHWOHE K G OJ¡ 1 1 1 1 1 S 1 1 | GHLP OQI WHOE KGWP PO H URPY O T « À m ¤ _ EGLQI h « ¤ _ ] h T g R KG « q m q _ E GLQI _ q _ ] r Y 1 1 1 P I GLIM PO OFIGHNG PRLP « À m ¤ _ ] « À m ¤ « À _T LQV PR I L UGQGHLK JHOJGHPM OW 1 1 QDEFGHIT g R R LKIO ROKVI W g G RLNG IDFPHL P OQ QIPGLV OW LVV P OQ k 1 S 1 S 1 1 1 1 ¤ ¿ ] ¤ ¿ LQV ¤ ¿ ] ¤ ¿ ¨ q ¿ ] q ¿ LQV q ¿ ] q ¿ ¨ W g G PL G ] q h T g G UGP q ¤ ¿ ] q q ¿ LQV q q ¿ ] q ¤ ¿T U N QU DI L g L M | 1 1 OW UGPP QU H V OW JLHGQPRGIGI g R G Q g G RLNG L E QDI I UQ Q WHOQPY 1 1 1 1 1 © $ ª _ q « q h q m q ¤ _ ¤ a h r q m ] _ q a q m q « ¤ a À ] _ q a ¤ h « ¤ _ c ] a h  Z _ q « q h q m q ¤ _ ¤ a h r q m ] _ q « ¤ h ¤ _ h q b ¤ a r q h _ ] a h à ? B » · = Ä @ Å ¶ · = @ B Q G L HGLK QDEFGH e I L JO QP OQ PRG HGLK K QGT g G LQ OQI VGH PI V IPLQ G WHOE 1 S 1 1 1 S S 1 1 1 S PRG OH U Q T g R R g G LKK PRG OW e LQV VGQOPG FM fe f Y 1 1 1 S S ¨ ¨ |y| |x| Æ Æ È Ç Ç É Æ Æ È Ç Ç É y x -4 -3-2 -10 1 2 3 4 UDHG h Y « k j R G LFIOKDPG NLKDG I PRG V IPLQ G WHOE PRG OH U Q 1 1 1 S 1 1 W PRG QDEFGH e I JOI P NGT PRGQ fe f I G t D L K PO PRG QDEFGH PIGKW f_ f ] _ Y W PRG QDE¡ 1 1 1 1 1 FGH I QGULP NGT ILM e ] q _ T PI V IPLQ G fe f WHOE PRG OH U Q I _Y COPG PRLP Q PR I LIG PRG 1 1 1 1 S 1 1 1 1 1 S LFIOKDPG NLKDG fe f ] _ I PRG QGULP NG OW e ] q _ T QLEGKM fe f ] f q _ f ] _ ] q q _ ] q e Á 1 1 IO W e I QGULP NGT fe f ] q e Y j R G KLIP G t D L K PM I L F P L g g L H V I Q G fe f I QGNGH QGU¡ 1 1 1 1 1 1 | 1 S 1 © LP NGT FDP g G QGGV PRG E QDI I UQ Q WHOQP OW e T I Q G e RLI LQOPRGH QGULP NG I UQ ¨ Q 1 1 1 1 1 S 1 1 1 a PÁ PR I g L M PRG P g O QGULP NG I UQI LQ GK ODP LQV g G UGP L JOI P NG ODP OE GY O g G LQ 1 1 1 1 S S 1 1 S S V G Q G PRG LFIOKDPG NLKDG OW L QDEFGH e L I k e W e Ê r 1 fe f ] u q e W e r 1 ËGHG LHG IOE G JHOJGHP GI OW PRG LFIOKDPG NLKDG PRLP ROKV WOH LKK HGLK Q D E F G H I k 1 ZDKG L E JKG f q f ] f f f q _ f ] f_ f T FG LDIG PRGM FOPR G t D L K _Y S f f ] f f À f f f q _ m f ] f q _ f À fm fT FG LDIG FOPR G t D L K b Y S f f _ f_ f _ ] ] T I Q G PRGM FOPR G t D L K Y 1 S f f q m f q m f m x x x x x x x x x x x x x x f ¤ f f f ¤ f f fa ¤ q m f fa f ¤ f q m fT I Q G h Y 1 S © # ª L fh q } _ f ] q h q } _ T I Q G h q } _ I QGULP NGY 1 S 1 1 } } } } _ q f_ q f_ q f_ q _ _ f _ f _ f F ] ] ] ] h Y x x } } } } _ q _ f _ q _ f f q _ q _ f f_ q _ f x x x x x x x x W Ê _T Q V f q h f Y © ( ª Q G Ê _T g G UGP Ê h Y O q h Ê r T QLEGKM q h I QOQ¡QGULP NGY j R D I T 1 S 1 1 f q h f ] q h Y fe Ì ¤ h f ] e Ì ¤ h T I Q G e Ì ¤ h I JOI P NG WOH LKK NLKDGI OW e Y 1 S 1 1 1 © % ª Y « fe f ¤ _ W e q m T LK DKLPG S S © ª x x x x ] « fe f ¤ _Y C O g T I Q G e I QGULP NGT fe f ] q e Á « fe f ¤ _ Q G « fe f ¤ _ I JOI P NGT 1 S 1 1 1 S 1 1 1 x x x x IOT « fe f ¤ _ ] « q e ¤ _ ] q « e ¤ _Y x x x x « QV LKK NLKDGI OW e ID R PRLP fe f ] m Y 1 S © Í ª ZG LKK PRLP fe f ] m I PRG V IPLQ G WHOE e PO PRG OH U Q T IO e LQ FG OQKM q m OH m T S 1 1 S 1 1 S QLEGKM e ] Î m Y QV LKK NLKDGI OW e ID R PRLP f_ e q « f ] Y 1 S © ¦ Ï ª ZG LKK PRLP f_ e q « f ] I PRG V IPLQ G WHOE _ e q « PO PRG OH U Q Á IO _ e q « LQ FG S 1 1 S 1 1 S OQKM q OH T QLEGKM _ e q « ] Î Y O T _ e q « ] q OH _ e q « ] T QLE GKM _ e ] q _ OH _ e ] h _ Y O e ] q h OH e ] b Y j R G IGP Ð e fe f _ Ñ OQPL QI LKK QDEFGHI OQ PRG HGLK K QG PRLP LHG KGII S 1 1 © ¦ ¦ ª x PRLQ _ DQ PI L g L M WHOE PRG OH U QY 1 1 1 x x Ò Ò Ô Ó Ó Õ Ò Ò Ô Ó Ó Õ -4-3 -2 -1 0 1 2 3 4 UDHG h Y b k j R G IGP e fe f _ Y 1 Ð Ñ x x x fe f Ö _ Ñ OQPL QI LKK QDEFGHI OQ PRG HGLK K QG PRLP LHG EOHG j R G IGP.
Load more

Recommended publications
  • Algebra/Geometry/Trigonometry App Samples
    Algebra/Geometry/Trigonometry App Samples Holt McDougal Algebra 1 HMH Fuse: Algebra 1- HMH Fuse is the first core K-12 education solution developed exclusively for the iPad. The portability of a complete classroom course on an iPad enables students to learn in the classroom, on the bus, or at home—anytime, anywhere—with engaging content that provides an individually-tailored learning experience. Students and educators using HMH Fuse: will benefit from: •Instructional videos that teach or re-teach all key concepts •Math Motion is a step-by-step interactive demonstration that displays the process to solve complex equations •Homework Help provides at-home support for intricate problems by providing hints for each step in the solution •Vocabulary support throughout with links to a complete glossary that includes audio definitions •Tips, hints, and links that enable students to acquire the help they need to understand the lessons every step of the way •Quizzes that assess student’s skills before they begin a concept and at strategic points throughout the chapters. Instant, automatic grading of quizzes lets students know exactly how they have performed •Immediate assessment results sent to teachers so they can better differentiate instruction. Sample- Cost is Free, Complete App price- $59.99 Holt McDougal HMH Fuse: Geometry- Following our popular HMH Fuse: Algebra 1 app, HMH Fuse: Geometry is the newest offering in the HMH Fuse series. HMH Fuse: Geometry will allow you a sneak peek at the future of mobile geometry curriculum and includes a FREE sample chapter. HMH Fuse is the first core K-12 education solution developed exclusively for the iPad.
    [Show full text]
  • Developing Creative Thinking in Mathematics: Trigonometry
    Secondary Mathematics Developing creative thinking in mathematics: trigonometry Teacher Education through School-based Support in India www.TESS-India.edu.in http://creativecommons.org/licenses/ TESS-India (Teacher Education through School-based Support) aims to improve the classroom practices of elementary and secondary teachers in India through the provision of Open Educational Resources (OERs) to support teachers in developing student-centred, participatory approaches. The TESS-India OERs provide teachers with a companion to the school textbook. They offer activities for teachers to try out in their classrooms with their students, together with case studies showing how other teachers have taught the topic and linked resources to support teachers in developing their lesson plans and subject knowledge. TESS-India OERs have been collaboratively written by Indian and international authors to address Indian curriculum and contexts and are available for online and print use (http://www.tess-india.edu.in/). The OERs are available in several versions, appropriate for each participating Indian state and users are invited to adapt and localise the OERs further to meet local needs and contexts. TESS-India is led by The Open University UK and funded by UK aid from the UK government. Video resources Some of the activities in this unit are accompanied by the following icon: . This indicates that you will find it helpful to view the TESS-India video resources for the specified pedagogic theme. The TESS-India video resources illustrate key pedagogic techniques in a range of classroom contexts in India. We hope they will inspire you to experiment with similar practices. They are intended to complement and enhance your experience of working through the text-based units, but are not integral to them should you be unable to access them.
    [Show full text]
  • Meaning and Understanding of School Mathematical Concepts by Secondary Students: the Study of Sine and Cosine
    EURASIA Journal of Mathematics, Science and Technology Education, 2019, 15(12), em1782 ISSN:1305-8223 (online) OPEN ACCESS Research Paper https://doi.org/10.29333/ejmste/110490 Meaning and Understanding of School Mathematical Concepts by Secondary Students: The Study of Sine and Cosine Enrique Martín-Fernández 1*, Juan Francisco Ruiz-Hidalgo 1, Luis Rico 1 1 Department of Mathematics Education, Faculty of Education, University of Granada, Campus Universitario Cartuja s/n, 18011, Granada, SPAIN Received 18 November 2018 ▪ Revised 5 April 2019 ▪ Accepted 12 April 2019 ABSTRACT Meaning and understanding are didactic notions appropriate to work on concept comprehension, curricular design, and knowledge assessment. This document aims to delve into the meaning of school mathematical concepts through their semantic analysis. This analysis is used to identify and establish the basic meaning of a mathematical concept and to value its understanding. To illustrate the study, we have chosen the trigonometric notions of sine and cosine of an angle. The work exemplifies some findings of an exploratory study carried out with high school students between 16 and 17 years of age; it collects the variety of emergent notions and elements related to the trigonometric concepts involved when answering on the categories of meaning which have been asked for. We gather the study data through a semantic questionnaire and analyze the responses using an established framework. The subjects provide a diversity of meanings, interpreted and structured by semantic categories. These meanings underline different understandings of the sine and cosine, according to the inferred themes, such as length, ratio, angle and the calculation of a magnitude.
    [Show full text]
  • Originally TRIGONOMETRY Was That Branch of Mathematics Concerned
    Originally TRIGONOMETRY was that branch of mathematics concerned with solving triangles using trigonometric ratios which were seen as properties of triangles rather than of angles. The word Trigonometry comes from the Greek words: Treis= three, Gonia=angle, and Metron=measure. Therefore, the word trigonometry means “measurement of triangles.” This unit will explore trigonometry as it relates to right triangles. Math 99 Intermediate Algebra Whatcom Community College revised by H Ypma Winter 2003 Introduction to Trigonometry Trigonometry is the branch of mathematics that studies the relationship between angles and sides of a triangle. Using “special trig functions”, you will be able to calculate the height of a tree from its shadow or the distance across a wide lake by walking and measuring along its shore. Let’s begin by working with a RIGHT TRIANGLE (one with a 90° angle in it) and using it to define those “special trig functions” we mentioned earlier. A First of all, we will use standard notation when labeling our right triangles. Capital letters will represent the vertices or angles; A, B, and C will be our standard letters. C will be used to represent the right angle. We will use lower case c b letters to represent the sides; a, b, and c will be the standard choices here. The side opposite the angle labeled A will be side “a”, the side opposite angle B will be “b” and the side opposite angle C will be “c”. C B a In a Right Triangle, the right angle will always measure 90° and the two other angles will add up to 90° .
    [Show full text]
  • Combinatorial Trigonometry (And a Method to DIE For)
    Combinatorial Trigonometry (and a method to DIE for) by Arthur T. Benjamin Harvey Mudd College MCSD Seminar Series National Institute of Standards and Technology January 11, 2008 Combinatorics Combinatorics n! = the number of ways to arrange 1, 2, ... n. Combinatorics n! = the number of ways to arrange 1, 2, ... n. n n! = k k!(n k)! ! " − = number of size k subsets of 1, 2, . , n { } = row n column k entry of Pascal’s triangle. Example: n = 4 4 4 4 4 4 0 1 2 3 4 ! "! "! "! "! " 1 4 6 4 1 counts subsets of 1, 2, 3, 4 { } size 0 1 2 3 4 ∅ 1 12 123 1234 2 13 124 3 14 134 4 23 234 24 34 Pascal’s Triangle Row 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 6 1 6 15 20 15 6 1 Pascal’s Triangle Row 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 6 6 6 6 6 6 6 6 0 1 2 3 4 5 6 ! "! "! "! "! "! "! " Patterns in Pascal’s Triangle Row Sum Row 0 1 1 1 1 1 2 2 1 2 1 4 3 1 3 3 1 8 4 1 4 6 4 1 16 5 1 5 10 10 5 1 32 6 1 6 15 20 15 6 1 64 6 6 6 6 6 6 6 + + + + + + = 64 = 26 0 1 2 3 4 5 6 ! " ! " ! " ! " ! " ! " ! " 6 6 = 26.
    [Show full text]
  • The Calculus of Trigonometric Functions the Calculus of Trigonometric Functions – a Guide for Teachers (Years 11–12)
    1 Supporting Australian Mathematics Project 2 3 4 5 6 12 11 7 10 8 9 A guide for teachers – Years 11 and 12 Calculus: Module 15 The calculus of trigonometric functions The calculus of trigonometric functions – A guide for teachers (Years 11–12) Principal author: Peter Brown, University of NSW Dr Michael Evans, AMSI Associate Professor David Hunt, University of NSW Dr Daniel Mathews, Monash University Editor: Dr Jane Pitkethly, La Trobe University Illustrations and web design: Catherine Tan, Michael Shaw Full bibliographic details are available from Education Services Australia. Published by Education Services Australia PO Box 177 Carlton South Vic 3053 Australia Tel: (03) 9207 9600 Fax: (03) 9910 9800 Email: [email protected] Website: www.esa.edu.au © 2013 Education Services Australia Ltd, except where indicated otherwise. You may copy, distribute and adapt this material free of charge for non-commercial educational purposes, provided you retain all copyright notices and acknowledgements. This publication is funded by the Australian Government Department of Education, Employment and Workplace Relations. Supporting Australian Mathematics Project Australian Mathematical Sciences Institute Building 161 The University of Melbourne VIC 3010 Email: [email protected] Website: www.amsi.org.au Assumed knowledge ..................................... 4 Motivation ........................................... 4 Content ............................................. 5 Review of radian measure ................................. 5 An important limit ....................................
    [Show full text]
  • Trigonometry/Calculus – Curriculum Map
    Trigonometry/Calculus – Curriculum Map Marking Connection to text CCSS Vocabulary Period Trigonometry with What are the State Standards students will be What vocabulary will support the Applications (1983) learning? students’ learning? Chapter 1 Angle Trigonometric Functions Vertex BASIC CONCEPTS Geometry Degrees Sec.1-1: Angles and Degree Similarity, right triangles, and trigonometry Minutes Measure Define trigonometric ratios and solve Seconds problems involving right triangles Initial Side THE TRIGONOMETRIC G.SRT.6 Understand that by similarity, side Terminal Side FUNCTIONS ratios in right triangles are properties of the Directed Angle Sec.1-2: Sine and Cosine angles in the triangle, leading to definitions of Standard Position trigonometric ratios for acute angles. Coterminal Angles Sec.1-3: Values of the Sine and Cosine Functions Sine Function G.SRT.7 Explain and use the relationship Cosine Function between the sine and cosine of complementary Sec.1-4: Other Trigonometric Significant Digit angles. st Functions Scientific Notation 1 Tangent Function Marking Sec.1-5: Solving Right G.SRT.8 Use trigonometric ratios and the Secant Period Triangles Pythagorean Theorem to solve right triangles in Cosecant applied problems. Cotangent Sec.1-6: Trigonometric Angle of Elevation Functions of Arbitrary Angles Angle of Depression Quadrantal Angles Reference Triangle Reference Angle Chapter 2 Circular Functions and Their Circles Radian Measure Graphs Find arc lengths and areas of sectors of Semi-Circle circles CIRCULAR FUNCTIONS
    [Show full text]
  • Drawing with Trigonometry Peter Elsea 1/22/12 1
    Drawing with Trigonometry Drawing with Trigonometry Trigonometry is one of those subjects that leaves students asking "What's in it for me?" After all, knowing a lot of stuff about ancient Greeks and how to prove theorems isn't much use to a lawyer or a musician. Graphic arts is another story, at least if you want to write your own programs. There are no theorems to prove, but some of the basic ideas are necessary all the time. This paper1 covers basic concepts and show how to put them to work. Cartesian coordinates. Trigonometry lives on graph paper: Figure 1. Cartesian space is measured with grid lines. Figure 1 shows the usual layout. Each line of the grid has a number, and locations on the grid are defined by the intersection of the two nearest grid lines. It's a lot like my saying I live at the corner of Laurel and Cleveland, or I'll meet you at third and Beach. Since this is math, we count to name the lines, and we count starting with 0. The intersection of 0 and 0 is the origin, and lines left of the origin or below it have negative numbers. Also since this is math, we assign letters to indicate horizontal or vertical lines: X for counting to the left and Y for counting up. The horizontal numbered 0 is called the X axis, and the vertical numbered 0 is the Y axis. Any point on the paper can then be defined with an X and Y value. If the point is between lines, it can have fractional values.
    [Show full text]
  • Mathematics.Pdf
    Algebra Algebra II w/ Trigonometry AP Calculus AB AP Calculus BC AP Statistics Calculus III College Algebra Geometry Linear Algebra Pre-Calculus Pre-Calculus Career Internship Program Mathematics 101 Mr. Collin Voigt, Division Chair TEL: SC: (708) 579-6580, NC: (708) 579-6410 FAX: (708) 579-6038 EMAIL: [email protected] Mr. Joseph Barker, Assistant Division Chair Ms. Annette Orrico, Assistant Division Chair TEL: SC (708) 579-6581, NC (708) 579-6412 TEL: SC: (708) 579-6583, NC: (708) 579-6411 FAX: (708) 579-6038 FAX: (708) 579-6038 EMAIL: [email protected] EMAIL: [email protected] Mathematics Department Philosophy Th e mathematics curriculum has been developed to help students value mathematics, become confi dent in their abilities to do mathematics, become mathematical problem solvers, and to communicate and reason mathemati- cally. Students, as a result of the high school mathematics experiences, should be able to model problems with the appropriate operations and equations, apply a variety of approaches and techniques to solve problems, understand the underlying mathematical features of problems, see the applicability of mathematical ideas to common and complex problems, use logical reasoning to present an argument, and employ technology to explore mathematical ideas and solve problems. Students who successfully completed Algebra (Accel) in Grade 7th or 8th grade will receive one unit of high school credit on a pass/fail basis. Th e high school credit will be awarded aft er successful completion of one year of mathematics while enrolled in high school. Mathematics 102 Mathematics Department Standards The LTHS Mathematics Department has adopted the following eight principles in conjunction with both the Illinois State Standards and the Common Core State Standards.
    [Show full text]
  • A Note on the History of Trigonometric Functions and Substitutions
    A NOTE ON THE HISTORY OF TRIGONOMETRIC FUNCTIONS AND SUBSTITUTIONS Jean-Pierre Merlet INRIA, BP 93, 06902 Sophia-Antipolis Cedex France Abstract: Trigonometric functions appear very frequently in mechanism kinematic equations (for example as soon a revolute joint is involved in the mechanism). Dealing with these functions is difficult and trigonometric substi- tutions are used to transform them into algebraic terms that can be handled more easily. We present briefly the origin of the trigonometric functions and of these substitutions. Keywords: trigonometric substitutions, robotics 1 INTRODUCTION When dealing with the kinematic equations of a mechanism involving revolute joints appears frequently terms involving the sine and cosine of the revolute joint angles. Hence when solving either the direct or inverse kinematic equa- tions one has to deal with such terms. But the problem is that the geometric structure of the manifold build on such terms is not known and that only nu- merical method can be used to solve such equations. A classical approach to deal with this problem is to transform these terms into algebraic terms using the following substitution: θ 2t 1 − t2 t = tan( ) ⇒ sin(θ)= cos(θ)= 2 1+ t2 1+ t2 This substitution allows one to transform an equation involving the sine and cosine of an angle into an algebraic equation. Such type of equation has a structure that has been well studied in the past and for which a large number properties has been established, such as the topology of the resulting man- ifold and its geometrical properties. Furthermore efficient formal/numerical methods are available for solving systems of algebraic equations.
    [Show full text]
  • COMPASS/ESL Cover&Intro 7/1/04 9:13 AM Page 1
    5281_COMPASS/ESL Cover&Intro 7/1/04 9:13 AM Page 1 COMPASS/ESL® Sample Test Questions— A Guide for Students and Parents Mathematics College Algebra Geometry Trigonometry An ACT Program for Educational Planning 5281_COMPASS/ESL Cover&Intro 6/25/04 12:20 PM Page 2 Note to Students Welcome to the COMPASS Sample Mathematics Test! You are about to look at some sample test questions as you prepare to take the actual COMPASS test. The examples in this booklet are similar to the kinds of test questions you are likely to see when you take the actual COMPASS test. Since this is a practice exercise, you will answer just a few questions and you won’t receive a real test score. The answer key follows the sample questions. Once you are ready to take the actual COMPASS test, you need to know that the test is computer delivered and untimed— that is, you may work at your own pace. After you complete the test, you can get a score report to help you make good choices when you register for college classes. We hope you benefit from these sample questions, and we wish you success as you pursue your education and career goals! Note to Parents The test questions in this sample set are similar to the kinds of test questions your son or daughter will encounter when they take the actual COMPASS test. Since these questions are only for practice, they do not produce a test score; students answer more questions on the actual test. The aim of this booklet is to give a sense of the kinds of questions examinees will face and their level of difficulty.
    [Show full text]
  • Trigonometry Math Help
    Trigonometry Math Help Learning Objective: • Recognize right triangles as having 2 sides and a hypotenuse, 2 acute angles; and a right angle. • Recognize and identify a specific reference angle along with its related opposite side, adjacent side, and hypotenuse. • Identify which trigonometric ratio should be used to solve for an unknown based on the information given. • Recognize the difference between the ratios that define sine, cosine, and tangent. • Understand the mathematical calculations required to solve for the unknown. Understanding the Meaning of Trigonometry Tri --- gono --- metry 3 --- sides --- measurement Trigonometry is the study of measurement of three-sided figures. More specifically, trigonometry is the study of the measurements of the sides and angles in triangles. Ancient Greek mathematicians discovered that if two different right triangles had two matching sides in the same ratio, the angle forming those two sides is equal. Example: 10cm 5cm 14 cm 7 cm A A height height ratio = ratio = hypotenuse hypotenuse 5cm 7cm ratio = ratio = 10cm 14cm 1 1 ratio = ratio = 2 2 A 30°=∠ A 30°=∠ Since both ratios are equal, the matching angles are also equal. The ratio defined in this example is called sine. There are two other basic ratios called cosine and tangent. Junior High Math Interactives Page 1 of 4 ©2006 Alberta Education (www.LearnAlberta.ca) Shape and Space / Trigonometry / Object Interactive / Math Help Definition of the Trigonometric Ratios: opposite adjacent opposite sinθ = cosθ = tanθ = hypotenuse hypotenuse adjacent Helpful Hints How to Chose the Appropriate Ratio: 1. Begin by selecting the acute angle from which the diagram is “viewed”. 2.
    [Show full text]