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WEB APPENDIX 1: AMERICAN NATIONAL STANDARDS OF INTEREST TO DESIGNERS, ARCHITECTS, AND DRAFTERS WEB SITES ANSI . . . . . . . . . . . . . . . . . . . . . . . . . . .www.ansi.org ASME . . . . . . . . . . . . . . . . . . . . . . . . . .www.asme.org TITLE OF STANDARD Abbreviations for Use on Drawings and Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y1.1—1999 American National Standard Drafting Practice: Metric Drawing Sheet Size and Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.1M—1995 Decimal Inch Drawing Sheet Size and Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.1—1995 Line Conventions and Lettering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.2M—1992 (R1998) Multi and Sectional View Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.3M—1994 (R1999) Pictorial Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.4—1989 (R1999) Revision of Engineering Drawings and Associated Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.35M—1992 Dimensioning and Tolerancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.5M—1994(R1999) Dimensioning and Tolerancing with Mathematical Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.5.1M—1994 (R1999) Certification of Geometric Dimensioning and Tolerancing Professionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.5.2—(1995) Screw Thread Representation, Engineering Drawing and Related Documentation Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.6—1978 (R1998) Engineering Drawing and Related Documentation Practices— Screw Thread Representation (Metric Supplement) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.6AM—1981 (R1998) Gears and Splines Gear Drawing Standards—Part 1, for Spur, Helical, Double Helical, and Rack . . . . . . . . . . . . . . . . .ASME Y14.7.1—1971 (R1998) Gear and Spline Drawing Standards—Part 2, Bevel and Hypoid Gears . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.7.2—1978 (R1999) Castings and Forgings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.8M—1989 (R1996) Engineering Drawing and Related Documentation Practices— Mechanical Spring Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.13M—1981 (R1998) Electrical and Electronics Diagrams (includes supplements ANSI Y14.15a—1971 and ANSI Y14.15b—1973) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.15—1966 (R1988) Fluid Power Systems and Products—Moving Parts Fluid Controls— Method of Diagramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME/(NFPA) T3.28.9R1—1989 Engineering Drawings and Related Documentation Practices—Optical Parts . . . . . . . . . . . . . . . . . . .ASME Y14.18M—1986 (R1998) Types and Applications of Engineering Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.24—1999 (R2000) Digital Representation for Communication of Product Definition Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .US PRO/IPO—100—1993 Chassis Frames—Passenger Car and Light Truck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.32.1M—1999 Parts Lists, Data Lists, and Index Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.34M—1989 Surface Texture Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.36—1978 (R1996) Graphic Symbols: Electrical Wiring and Layout Diagrams Used in Architecture and Building Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.9—1972 (R1989) Plumbing Fixtures for Diagrams Used in Architecture and Building Construction . . . . . . . . . . . . . . . . . .ANSI Y32.4—1977 (R1999) Railroad Maps and Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.7—1972 (R1994) Fluid Power Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.10—1967 (R1999) Process Flow Diagrams in the Petroleum and Chemical Industries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.11—1961 (R1998) Mechanical and Acoustical Elements as Used in Schematic Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.18—1972 (R1998) Pipe Fittings, Valves, and Piping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI/ASME Y32.2.3—1949 (R1999) Heating, Ventilating, and Air Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.2.4—1949 (R1998) Heat-Power Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.2.6M—1950 (R1999) Welding, Brazing, and Nondestructive Examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI/AWS A2.4—1993 Letter Symbols: Glossary of Terms Concerning Letter Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y10.1—1972 (R1988) Quantities Used in Electrical Science and Electrical Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI/IEEE 280—1985 (R1992) Letter Symbols and Abbreviations for Quantities Used in Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI/ASME Y10.11—1984 Chemical Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y10.12—1955 (R1988) Illuminating Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y10.18—1967 (R1977) Mathematical Signs and Symbols for Use in Physical Sciences and Technology . . . . . . . . . . . . . . . . . . . . . . .ANSI/IEEE 260.3—1993 Engineering Drawing Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.100M—1998 Engineering Drawings and Associated Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.35M—1997 APPENDIX 1 1

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WEB APPENDIX 1: AMERICAN NATIONAL STANDARDS OF INTEREST TO DESIGNERS,ARCHITECTS, AND DRAFTERS

WEB SITESANSI . . . . . . . . . . . . . . . . . . . . . . . . . . .www.ansi.orgASME . . . . . . . . . . . . . . . . . . . . . . . . . .www.asme.org

TITLE OF STANDARDAbbreviations for Use on Drawings and Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y1.1—1999American National Standard Drafting Practice:

Metric Drawing Sheet Size and Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.1M—1995Decimal Inch Drawing Sheet Size and Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.1—1995Line Conventions and Lettering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.2M—1992 (R1998)Multi and Sectional View Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.3M—1994 (R1999)Pictorial Drawing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.4—1989 (R1999)Revision of Engineering Drawings and Associated Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.35M—1992Dimensioning and Tolerancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.5M—1994(R1999)Dimensioning and Tolerancing with Mathematical Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.5.1M—1994 (R1999)Certification of Geometric Dimensioning and Tolerancing Professionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.5.2—(1995)Screw Thread Representation, Engineering Drawing and Related

Documentation Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.6—1978 (R1998)Engineering Drawing and Related Documentation Practices—

Screw Thread Representation (Metric Supplement) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.6AM—1981 (R1998)Gears and Splines

Gear Drawing Standards—Part 1, for Spur, Helical, Double Helical, and Rack . . . . . . . . . . . . . . . . .ASME Y14.7.1—1971 (R1998)Gear and Spline Drawing Standards—Part 2, Bevel and Hypoid Gears . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.7.2—1978 (R1999)

Castings and Forgings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.8M—1989 (R1996)Engineering Drawing and Related Documentation Practices—

Mechanical Spring Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.13M—1981 (R1998)Electrical and Electronics Diagrams (includes supplements ANSI Y14.15a—1971

and ANSI Y14.15b—1973) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.15—1966 (R1988)Fluid Power Systems and Products—Moving Parts Fluid Controls—

Method of Diagramming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME/(NFPA) T3.28.9R1—1989Engineering Drawings and Related Documentation Practices—Optical Parts . . . . . . . . . . . . . . . . . . .ASME Y14.18M—1986 (R1998)Types and Applications of Engineering Drawings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.24—1999 (R2000)Digital Representation for Communication of Product Definition Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .US PRO/IPO—100—1993Chassis Frames—Passenger Car and Light Truck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.32.1M—1999Parts Lists, Data Lists, and Index Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.34M—1989Surface Texture Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.36—1978 (R1996)

Graphic Symbols:Electrical Wiring and Layout Diagrams Used in Architecture

and Building Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.9—1972 (R1989)Plumbing Fixtures for Diagrams Used in Architecture and Building Construction . . . . . . . . . . . . . . . . . .ANSI Y32.4—1977 (R1999)Railroad Maps and Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.7—1972 (R1994)Fluid Power Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.10—1967 (R1999)Process Flow Diagrams in the Petroleum and Chemical Industries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.11—1961 (R1998)Mechanical and Acoustical Elements as Used in Schematic Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.18—1972 (R1998)Pipe Fittings, Valves, and Piping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI/ASME Y32.2.3—1949 (R1999)Heating, Ventilating, and Air Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.2.4—1949 (R1998)Heat-Power Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y32.2.6M—1950 (R1999)Welding, Brazing, and Nondestructive Examination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI/AWS A2.4—1993

Letter Symbols:Glossary of Terms Concerning Letter Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y10.1—1972 (R1988)Quantities Used in Electrical Science and Electrical Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI/IEEE 280—1985 (R1992)Letter Symbols and Abbreviations for Quantities Used in Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI/ASME Y10.11—1984Chemical Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y10.12—1955 (R1988)Illuminating Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ANSI Y10.18—1967 (R1977)Mathematical Signs and Symbols for Use in Physical Sciences and Technology . . . . . . . . . . . . . . . . . . . . . . .ANSI/IEEE 260.3—1993

Engineering Drawing Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.100M—1998Engineering Drawings and Associated Documents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .ASME Y14.35M—1997

APPENDIX 1 ■ 1

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WEB APPENDIX 2: DIMENSIONING AND TOLERANCING SYMBOLS

2 ■ Appendix 2

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adhesive bonding . . . . . . . . . . . . . . . . . . . . . . . . . . . . ABDarc welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AW

atomic hydrogen welding . . . . . . . . . . . . . . . . . . . AHWbare metal arc welding . . . . . . . . . . . . . . . . . . . . BMAWcarbon arc welding . . . . . . . . . . . . . . . . . . . . . . . . CAW

gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CAW-Gshielded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CAW-Stwin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CAW-T

electrogas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EGWflux cored arc welding . . . . . . . . . . . . . . . . . . . . . FCAWgas metal arc welding . . . . . . . . . . . . . . . . . . . . . GMAW

pulsed arc . . . . . . . . . . . . . . . . . . . . . . . . . . GMAW-Pshort circuiting arc. . . . . . . . . . . . . . . . . . . . GMAW-S

gas tungsten arc welding . . . . . . . . . . . . . . . . . . . GTAWpulsed arc . . . . . . . . . . . . . . . . . . . . . . . . . . . GTAW-P

plasma arc welding. . . . . . . . . . . . . . . . . . . . . . . . . PAWshielded metal arc welding . . . . . . . . . . . . . . . . . SMAWstud arc welding. . . . . . . . . . . . . . . . . . . . . . . . . . . . SWsubmerged arc welding. . . . . . . . . . . . . . . . . . . . . . SAW

series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SAW-Sbrazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B

arc brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ABblock brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BBcarbon arc brazing . . . . . . . . . . . . . . . . . . . . . . . . . CABdiffusion brazing. . . . . . . . . . . . . . . . . . . . . . . . . . . DFBdip brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DBflow brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FLBfurnace brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FBinduction brazing. . . . . . . . . . . . . . . . . . . . . . . . . . . . IBinfrared brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . . IRBresistance brazing . . . . . . . . . . . . . . . . . . . . . . . . . . . RBtorch brazing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TB

other welding processeselectron beam welding. . . . . . . . . . . . . . . . . . . . . . EBW

high vacuum . . . . . . . . . . . . . . . . . . . . . . . . EBW-HVmedium vacuum . . . . . . . . . . . . . . . . . . . . . EBW-MVnonvacuum . . . . . . . . . . . . . . . . . . . . . . . . . EBW-NV

electroslag welding. . . . . . . . . . . . . . . . . . . . . . . . . ESWflow welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . FLOWinduction welding. . . . . . . . . . . . . . . . . . . . . . . . . . . IWlaser beam welding . . . . . . . . . . . . . . . . . . . . . . . . LBWthermit welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . TW

oxyfuel gas welding . . . . . . . . . . . . . . . . . . . . . . . . . OFWair acetylene welding . . . . . . . . . . . . . . . . . . . . . . . AAWoxyacetylene welding . . . . . . . . . . . . . . . . . . . . . . OAWoxyhydrogen welding . . . . . . . . . . . . . . . . . . . . . . OHWpressure gas welding . . . . . . . . . . . . . . . . . . . . . . . PGW

resistance welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . RWflash welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FWpercussion welding . . . . . . . . . . . . . . . . . . . . . . . . PEWprojection welding . . . . . . . . . . . . . . . . . . . . . . . . . . PWresistance seam welding. . . . . . . . . . . . . . . . . . . . RSEW

high frequency . . . . . . . . . . . . . . . . . . . . . . RSEW-HFinduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . RSEW-I

resistance spot welding . . . . . . . . . . . . . . . . . . . . . RSWupset welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UW

high frequency. . . . . . . . . . . . . . . . . . . . . . . . UW-HFinduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UW-I

soldering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sdip soldering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DSfurnace soldering. . . . . . . . . . . . . . . . . . . . . . . . . . . . FSinduction soldering . . . . . . . . . . . . . . . . . . . . . . . . . . ISinfrared soldering. . . . . . . . . . . . . . . . . . . . . . . . . . . IRSiron soldering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . INSresistance soldering . . . . . . . . . . . . . . . . . . . . . . . . . . RStorch soldering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TSwave soldering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . WS

solid-state welding . . . . . . . . . . . . . . . . . . . . . . . . . . . SSWcoextrusion welding . . . . . . . . . . . . . . . . . . . . . . . CEWcold welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CWdiffusion welding . . . . . . . . . . . . . . . . . . . . . . . . . DFWexplosion welding . . . . . . . . . . . . . . . . . . . . . . . . . EXWforge welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . FOWfriction welding . . . . . . . . . . . . . . . . . . . . . . . . . . . FRWhot pressure welding . . . . . . . . . . . . . . . . . . . . . . . HPWroll welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ROWultrasonic welding . . . . . . . . . . . . . . . . . . . . . . . . . USW

thermal cutting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TCarc cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AC

air carbon arc cutting . . . . . . . . . . . . . . . . . . . . . AACcarbon arc cutting . . . . . . . . . . . . . . . . . . . . . . . CACgas metal arc cutting . . . . . . . . . . . . . . . . . . . . GMACgas tungsten arc cutting . . . . . . . . . . . . . . . . . . GTACmetal arc cutting . . . . . . . . . . . . . . . . . . . . . . . . MACplasma arc cutting. . . . . . . . . . . . . . . . . . . . . . . . PACshielded metal arc cutting . . . . . . . . . . . . . . . . SMAC

electron beam cutting . . . . . . . . . . . . . . . . . . . . . . . EBClaser beam cutting. . . . . . . . . . . . . . . . . . . . . . . . . . LBC

air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LBC-Aevaporative . . . . . . . . . . . . . . . . . . . . . . . . . . LBC-EVinert gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LBC-IGoxygen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LBC-O

oxygen cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OCchemical flux cutting . . . . . . . . . . . . . . . . . . . . . FOCmetal powder cutting . . . . . . . . . . . . . . . . . . . . . POCoxyfuel gas cutting . . . . . . . . . . . . . . . . . . . . . . . OFC

oxyacetylene cutting . . . . . . . . . . . . . . . . . . OFC-Aoxyhydrogen cutting. . . . . . . . . . . . . . . . . . OFC-Hoxynatural gas cutting . . . . . . . . . . . . . . . . OFC-Noxypropane cutting . . . . . . . . . . . . . . . . . . . OFC-P

oxygen arc cutting . . . . . . . . . . . . . . . . . . . . . . . AOCoxygen lance cutting . . . . . . . . . . . . . . . . . . . . . LOC

thermal spraying . . . . . . . . . . . . . . . . . . . . . . . . . . . THSParc spraying. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ASPflame spraying . . . . . . . . . . . . . . . . . . . . . . . . . . . . FLSPplasma spraying. . . . . . . . . . . . . . . . . . . . . . . . . . . . PSP

APPENDIX 3 ■ 3

WEB APPENDIX 3: DESIGNATION OF WELDING AND ALLIED PROCESSES BY LETTERS

Welding and Letter Welding and Letter Allied Processes Designation Allied Processes Designation

Page 4: Simbología planos

WEB APPENDIX 4: SYMBOLS FOR PIPE FITTINGS AND VALVES

4 ■ Appendix 4

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Common Fractions

To add and subtract common fractions, the denominators (bot-toms) must be the same.

Normally, fractions are reduced to lowest form by dividingboth the numerator (top) and denominator by the same num-ber. Dividing by 4 reduces the previous fraction.

To multiply fractions, the denominators do not have to bethe same. Simply multiply the numerators and denominatorsseparately.

To divide fractions, flip the second fraction upside down(invert the divisor), then multiply.

When multiplying and dividing mixed numbers, it is neces-sary to convert them to fractions first. This is done by multi-plying the whole number by the denominator of the fractionand adding that to the top of the fraction for a new numerator.

For example:

The final answer was obtained by dividing the 99 by the 12with long division.

Decimal Fractions

To convert a common fraction to a decimal fraction, divide thenumerator by the denominator. Dividing 3 by 8 shows that

To convert a decimal to common fraction form, place thedecimal number without the decimal point on top and place itsplace value, given by this scheme, on the bottom of the fraction.

38

375= .

2 14

3 23

94

113

9912

8 14

× = × = =

6 38

6 8 38

518

= × +

=

14

23

14

32

38

÷ = × =

14

23

212

16

× = =

416

14

=

1

163

164

16+ =

Examples are:

With the advent of computers and calculators, it has becomeimportant to know how to round decimal answers properly.Look one place to the right of where the decimal is to berounded. If that digit is 5 or more, increase the value of thedigit to the left of it by one; if it is 4 or less, do not increase theleft digit. This concept is best grasped by viewing examples.

Rounding to the nearest hundredth:

.275 becomes .28 (due to the 5 one place to the right ofthe hundredth position)

.273 become .27 (the 3 is “4 or less”)

.27499 also becomes .27 (the 9s have no effect on round-ing to hundredths)

.279 becomes .28 (the 9 is “5 or more”)

Rounding to the nearest tenth:

.275 becomes .3

.274 also becomes .3 (look only to the 7)

.249 becomes .2 (look only to the 4)

.05 becomes .1

.04 becomes .0

.729 becomes .7

Percentages

Percents are hundredths:

Percentages are easiest to calculate in decimal form. To con-vert a percentage to a decimal, move the decimal point twoplaces to the left. Examples follow.

5% = .0550% = .50

3.75% = .0375Success in working percentage problems requires correctly

identifying three pieces of the problem called the part, the base,and the rate. The part is a portion of any whole amount, thebase is the whole amount, and the rate is the number with thepercent sign (%).

To find the part:

1. Convert the rate to decimal form.

2. Multiply the rate by the base.

5 5100

120

50 50100

12

%

%

= =

= =

.

.

.

375 3751000

38

014 141000

7500

6 75 6 75100

6 34

= =

= =

= =

APPENDIX 5 ■ 5

WEB APPENDIX 5: MATH INSTRUCTIONARITHMETIC

0 . 1 4 1 5 9 2tenthhundredththousandthten thousandthhundred thousandthmillionth

Page 6: Simbología planos

Example 1: What is 15% of 233?In this example the rate is 15% and 233 is the whole amount

or base. The answer is 34.95 because 15% × 233 = .15 × 233 = 34.95.

Example 2: What is 5.25% of 1000?The answer is 52.5 because .0525 × 1000 = 52.5.

Example 3: What is 125% of 500?The answer is 625 because 1.25 × 500 = 625.

To find the base:

1. Convert the rate to decimal form.

2. Divide the part by the rate.

Example 4: 10% of what number is 375?This problem requires realizing that 375 is a portion of some

unknown whole amount. So you know the rate and part andare seeking the base.

The answer is 3,750 because 375 ÷ .10 = 3,750.

Example 5: An old copy machine is known to waste 2% of thecopies it makes. How many copies were run if 14 were wasted?

The wasted copies are the part and the entire run is the base,so the answer is 14 ÷ .02 = 700 copies run.

To find the rate:

1. Divide the part by the base.

2. Convert the answer to percent by moving the decimalpoint two places to the right.

Example 6: What percent of 500 is 125?This type of problem requires identifying which number is

the base. From the wording, the whole amount is 500 so that isthe base, and 125 is a portion of this whole amount, making itthe part. Often, the base is the number immediately followingthe word of. The answer to the problem is 25% because 125 ÷ 500 = .25, which equals 25%.

Example 7: A machine cutting tool has a useful life of 110 hours.If the tool is used 50 hours, what percentage of its useful life isleft?

In this problem first find the useful life left in hours: 110 – 50 = 60. So this problem is the same as asking what per-cent of 110 is 60. The solution is 54.5% because 60 ÷ 110 =.545 (after rounding), then .545 = 54.5%.

Example 8: $200 is what percent of $50?The answer is 400% because 200 ÷ 50 = 4 = 400%. The

moral of this example is that the base is not always the largernumber.

Powers and Roots

A power is a small, raised number and stands for repeated multiplication.

Example 1: 53 = 5 × 5 × 5 = 125. This third power is called acube. The example is read, “5 cubed equals 125.”

Example 2: 72 = 7 × 7 = 49. The second power is the square, andthis example is read, “7 squared equals 49.”

A root is the reverse of a power.

Example 3: 3√125 = 5. This cube root of 125 equals 5 because 5 cubed is 125.

Example 4: The square root is the most common root. Thesquare root of 49 may be written

2√49, but the index number 2

is usually suppressed for square root. Write √49 = 7. It equals 7because 7 squared is 49.

Here are other examples of powers and roots.

25 = 32 62 = 36 3√8 = 2 √16 = 4

All scientific calculators have buttons or sequences of but-tons that make finding powers and roots very easy. The book-let that comes with the calculator should be consulted to learnhow to do this.

ALGEBRASigned Numbers

The sign of a number is found immediately to the left of a num-ber. If there is no sign, this means the same as if there is a plussign. For example, in the expression “4 – 7” the 4 is a positivenumber and the 7 is a negative number. Multiplication of twonumbers is indicated by parentheses around one or both num-bers and no sign between the numbers: (–3)(–6) is multiplica-tion, but (–3) + (–6) is not multiplication. Here are other exam-ples of multiplication:

(2)(–3) = –6–7(8) = –56

(–5)(–6) = 307(4) = 28

As these examples show, the rules for signs when multiply-ing are:

1. If signs are the same, the answer is positive.

2. If signs are different, the answer is negative.

Multiplication is also indicated by a raised dot (not to beconfused with a decimal point).

–4 ⋅ –3 = 12The sign rules for division are identical to multiplication. For

example, –9 ÷ 3 = –3. Often in algebra, division is indicated asa fraction. The previous division example might be expressed as

The rules for combining (adding and subtracting) signednumbers are:

1. If signs are the same, add the numbers and give the answerthe common sign.

2. If signs are different, subtract the two numbers and givethe answer the sign of the bigger.

Example 1: –1 +3This is not a multiplication problem because there are no

parentheses or raised dots; it is a combining problem. The 1 isnegative and the 3 is positive, so their signs are different, which

−93

6 ■ Appendix 5

Page 7: Simbología planos

calls for subtraction: 1 from 3 leaves 2. The 3 is bigger than the1 so the answer gets the positive sign of the 3. Thus, –1 +3 = 2.

Example 2: –4 –6This is also a combining problem. Both numbers are nega-

tive so their signs are the same. Add 4 and 6 and get 10. Givethe answer the common negative sign, so –4 –6 = –10. It doesnot matter whether you interpret this problem as “–4 add –6”or “–4 subtract +6,” but the easiest interpretation is “combine–4 and –6.” Here are further examples of combining problems:

+7 –5 = +2+5 –7 = –2–3 –1 = –4–4 +3 = –1

Sometimes parentheses do occur in combining problemsinvolving equations. Such problems require parentheses to firstbe removed by using the rules of multiplication on the signsaround the left parenthesis of each pair. The following exampleillustrates this.

Example 3: (–3) + (–6)The (–3) has no sign in front so a positive sign can be

attached: +(–3). Now the signs around the left parenthesis sym-bol are + and –. These signs are different, and applying the mul-tiplication sign rule for them (different signs, answer is nega-tive) resolves them into a single negative (–) sign. Likewise, thesigns around the left parenthesis of the 6 resolves them into asingle negative (–) sign. So the problem (–3) + (–6) is the sameas –3 –6. Using the combining rules on this, –3 –6 = –9.

Example 4: –4 – (–5) = –4 +5 = 1

Example 5: 6 – (7) = +6 – (+7) = +6 –7 = –1Sometimes technical publications and applications omit the

parentheses and show just the double signs. Solve such com-bining problems in the same way:

–4 – –5 = –4 +5 = 1Here are further examples of combining problems:

–4 – (–4) = –4 +4 = 03 – (–3) = 3 + 3 = 6

2 + (–7) – (8) = 2 – 7 –8 = –5 –8 = –13

Evaluating Expressions

When using formulas, the first step is to replace (substitute)the letters with numbers. The resulting expression is thensolved by applying the rules for signed numbers.

Example 1 (temperature): Find F if C = –10 using the formula

F = 1.8 C + 32Replacing the letters with numbers gives F = (1.8) (–10) + 32.In algebra, when a letter is next to a number and there is no

+ or – sign, multiplication is indicated. This formula calls for Cto be multiplied by 1.8: (1.8) (–10) = –18. So we now have F = –18 + 32, which gives F = 14.

Since most formulas involve a mixture of addition, multipli-cation, and other math operations, it is necessary to follow theworldwide order of operations when evaluating expressions:

1. Work within parentheses first.

2. Then do powers and roots.

3. Then do multiplication and division.

4. Finally, do addition and subtraction.

Example 2: (5 – 1)2 – 3 = (4)2 – 3 = 16 – 3 = 13

Example 3: 17 – 8 ÷ 4 = 17 – 2 = 15 (It is tempting to subtractthe 8 from the 17, but the division must be done first.)

Solving Simple Equations

There are basically two operations that solve simple equations.One operation involves combining, the other involves multi-plying. When a letter (unknown) has a number combined withit, move that number to the other side of the equal sign but alsochange its sign.

Example 1: x + 5 = 7Moving the +5 to the right side of the equation and chang-

ing its sign to –5 gives x = 7 – 5. Evaluating x = 2.

Example 2: 10 = –4 + y10 +4 = +y

or just: 10 +4 = yso: 14 = yor: y = 14

Example 3: 3 + z = 2z = 2 – 3z = –1

Example 4: Solve for x.

x + abc = zx = z – abc

When a letter is being multiplied by a number, divide bothsides of the equation by that number.

Example 5:

Example 6:

Example 7:

Conversely, if the problem has division, multiply.

Example 8: x

x

x

26

22

6 2

12

=

( )

= ( )( )=

Ax B

Ax

A

B

A

xB

A

=

=

=

− =−−

=−

= −

2 10

2

2

10

25

y

y

y

5 15

5

5

15

53

x

x

x

=

=

=

APPENDIX 5 ■ 7

Page 8: Simbología planos

The solution of larger simple equations involves applyingthese two basic operations more than once.

Example 9:

Example 10:

To solve an equation with a letter squared, take the squareroot of both sides of the equation.

Example 11:

Conversely, to solve an equation with the square root of aletter, square both sides of the equation.

Example 12:

Ratio and Proportion

One very common and useful type of equation is called a pro-portion. Each side of a proportion is a fraction or ratio. To solvea proportion by cross-multiplying, multiply the top of one ratioby the bottom of the other and set them equal.

Example 1:

An older way to express this example is X:6::15:5. The x and5 were the extremes and the 6 and 15 the means.

Example 2: On a drawing, the scale is 1:150. What actual lengthdoes a 3.25'' line of the drawing represent?

Set up a proportion.

1150

3 25

1 150 3 25487 5 40 7 1

2

''''

. ''

.. '' ' ''

=

( ) = ( )( )= −

xxx or

x

x

x

x

x

6

15

55 6 15

5 90

5

5

90

518

=

( )( ) = ( )( )=

=

=

y

y

y

=

( ) =

=

8

8

64

22

because also equals7 7 49− −⋅( )

x

x

x and x

2

2

49

49

7 7

=

== = −

− − =− = +− =−

=−

= −

2 9 11

2 11 9

2 20

2

2

20

210

y

y

y

y

y

5 7 22

5 22 7

5 15

5

5

15

53

x

x

x

x

x

+ == −=

=

=

Example 3: A cylindrical container holds 1,000 gallons of oilwhen filled to a depth of 8 feet. How many gallons are therewhen the depth is 31⁄2 feet?

FIGURE 1 ■ A cylindrical container.

Set up a proportion.

GEOMETRYTwo-Dimensional Figures

This section contains formulas for the perimeter (P) and areaof common geometric figures the drafter may encounter.Perimeter is the straight-line (linear) distance around a fig-ure, and area is the number of square units that fit within afigure.

Right TriangleA + B = 90°x2 + y2 = r2

P = x + y + r

FIGURE 2 ■ Right triangle.

Area1

2xy=

10008

8 1000 3 58 3500

437 5

.

.

gallonsfeet

x gallons3.5 feet

xxx gallons

=

( )( ) = ( )( )==

8 ■ Appendix 5

Page 9: Simbología planos

General TriangleA + B + C = 180°P = a + b + c

RectangleP = 2a + 2bArea = ab

FIGURE 4 ■ Rectangle.

ParallelogramA + B = 180°P = 2a + 2bArea = bh

FIGURE 5 ■ Parallelogram.

FIGURE 6 ■ Rhombus.

RhombusP = 4a

Note: The letters and p and q represent diagonal distances.p2 + q2 = 4a2

Area1

2pq=

d a b2 2= +

Area1

2bh

Area s s a s b s c where s1

2a b c

=

= −( ) −( ) −( ) = + +( )

CircleR = radiusD = diameterπ = 3.14159 . . .C = the circle’s perimeter, or circumferenceC = πDArea = πr2

Circle Inscribed within a Right Triangle

FIGURE 8 ■ Circle within a right triangle.

Circle Inscribed within a General Triangle

FIGURE 9 ■ Circle within a general triangle.

Rs a s b s c

s where s

1

2a b c=

−( ) −( ) −( )= + +( )

Rab

a b c=

+ +

AreaD

4

2

= π

APPENDIX 5 ■ 9

FIGURE 3 ■ General triangle.

FIGURE 7 ■ Circle.

Page 10: Simbología planos

Circle Circumscribed around a Right Triangle

Note: The letter c is the diameter of the circle as well as thehypotenuse of the right triangle.

FIGURE 10 ■ Circle around a right triangle.

Circle Circumscribed around a General Triangle

FIGURE 11 ■ Circle around a general triangle.

Angle Relationship for the Genera1 Triangle Inscribed within a Circle

FIGURE 12 ■ Angles within an inscribed triangle.

B1

2D=

Ra

2 sin A

b

2 sin B

c

2 sin C= = =

R1

2c=

Regular Polygonsn = number of sidesP = nf

FIGURE 13 ■ Regular polygon.

Arc of a Circles = length of the arc of the circle (arc length)Note: θ must be in radians.

FIGURE 14 ■ Arc of a circle.

Ellipse

(The perimeter formula is approximate.)Area = πxy(The area formula is exact.)

FIGURE 15 ■ Ellipse.

P 2x y

2

2 2

=+

π

s = R

F 2R sin2

g R cos2

h R g

θθ

θ

=

=

= −

θ = −

°

= °

= °

n 2

n180

f 2R sin 180

n

Area1

2nR sin

360

n

df

2 tan 180

n

2

10 ■ Appendix 5

Page 11: Simbología planos

Three-Dimensional Figures

This section shows formulas for total surface area (S) in squareunits and volume (V) in cubic units for common shapes.

Rectangular SolidS = 2(wh + lw + lh)V = lwh

FIGURE 16 ■ Rectangular solid.

CylinderS = 2πR2 + 2πRhV = πR2h

FIGURE 17 ■ Cylinder.

Right Circular Cone

FIGURE 18 ■ Right circular cone.

Sphere

FIGURE 19 ■ Sphere.

S RS D where D = 2R

V4

3R

2

3

==

=

42

ππ

π

,

S R RL

L R h

V1

3R h

2

2 2

2

= +

= +

=

π π

π

TRIGONOMETRYRight Triangle Trigonometry

Trigonometry is based upon the lengths of sides and the anglesof right triangles. It is best to treat the trigonometric functiondefinitions as simply working formulas involving two sides andan angle of the right triangle of Figure 20 and to use the for-mula that involves the information in the problem. Here are thedefinitions of the trig functions: In the right triangle, thelongest side (r) is called the hypotenuse. The side (y) is calledthe opposite side, because it is opposite to angle (A). Side (x) iscalled the adjacent side.

The first formula says that dividing the length of side y byhypotenuse r gives a number called the sine of angle A. (Sine isabbreviated “sin” but still pronounced sine.)

FIGURE 20 ■ Right triangle used for trigonometry definitions.

Example 1: A very common right triangle is the 3-4-5 right trian-gle. In Figure 21, find the sine of angle A.

Using the sine formula:

FIGURE 21 ■ 3-4-5 right triangle.

Sin Ayr

Sin A 35

=

= = .6

Sin Ayr

Cos A xr

Tan Ayx

=

=

=

Sin Ayr

Cos A xr

Tan Ayx

=

=

=

APPENDIX 5 ■ 11

Page 12: Simbología planos

The usefulness of trigonometry is that it provides a connec-tion between the lengths of sides and measurement of angles.Knowing, for example, that sin A = .6, a calculator can be usedto find angle A. For most calculators, entering .6 then pushingINV and SIN (or 2ND and SIN) gives angle A as 36.9°.

Similarly, cosine is the adjacent side divided by thehypotenuse, and tangent is the opposite side divided by theadjacent side. An additional formula used to solve right trian-gles involves only the sides. It is the Pythagorean theorem:

x2 + y2 = r2

Example 2: Find the diagonal distance in Figure 22 for a rec-tangular plate. Using the formula:

x2 + y2 = r2

72 + 102 = r2

49 + 100 = r2

149 = r2

FIGURE 22 ■ Rectangular plate.

Example 3: Find the length of side x for the right triangle in Fig-ure 23.

Because you know angle A and side y, use the tangent for-mula; only tangent involves the unknown side x and theknown angle and side.

FIGURE 23 ■ Right triangle.

Angle Conversion and Arc Length

One degree equals 60 minutes of arc: 1° = 60'One minute equals 60 seconds of arc: 1' = 60''

Therefore, 1° = 3600''

Example 1: Convert 4°50'35'' to decimal degrees (nearesthundredth).

tan A =y

x

tan 4012''

x

.839112''

xx

x12''

.8391x 14.3''

° =

=

=

=

=

. ' '8391 12

r = =149 12 21. ''

The answer is

Example 2: Convert 10.268° to degrees, minutes, and seconds.Working with the decimal fraction of the degrees, .268 × 60

= 16.08 for 16 whole minutes. Then, working with the decimalfraction of minutes, .08 × 60 = 4.8 or (rounding off) 5 wholeseconds. The complete answer becomes 10°16'5''.

Here are additional conversion facts:

Degrees and Radians:

180° = π radians (where π = 3.14159 . . .)1 radian = about 57.3°

1° = about .01745 radian

Example 3: Convert 30° to radians (nearest thousandth). Theanswer is 30 × .01745 = .5235 or .524 radian.

Example 4: Convert 2 radians to degrees (nearest tenth). Theanswer is 2 × 57.3 = 114.6°.

Vectors

A vector is a directed line segment, or arrow, with two attrib-utes: length (magnitude) and angle (direction). Points in theplane may be specified by their x and y coordinates or by mag-nitude and direction of a vector with its tail at the origin and itshead at the point in the plane as shown in Figure 24.

Vector notation and equations conform to the definitions ofthe trig functions except that Greek letter θ (theta) is oftenused for the angle A. Here are the conversion formulas:

Polar to Rectangular (vector form to x-y form):

x = r cos θy = r sin θ

The x and y are also called vector components.

FIGURE 24 ■ Vector.

4 5060

353600

4 8333 0097

4 843

+ + = + +

= °

. .

. or 4.84

12 ■ Appendix 5

Page 13: Simbología planos

Example 1: Find the components of the vector shown in Figure 25.Using the conversion formulas with r = 10 and θ = 30°,

x = (10)(cos 30°) = (10)(.8660) = 8.66 lb, andy = (10)(sin 30°) = (10)(.5) = 5 lb

FIGURE 25 ■ Vector in polar form.

Rectangular to Polar (x-y form to vector form):

Example 2: Convert the coordinates (7,13) in Figure 26 to polarform.

Using the conversion formulas with x = 7 and y = 13,

FIGURE 26 ■ Vector in rectangular form.

Slope of a Line

The slope-intercept form of the equation of a straight line is: y = mx + b. Letter m is the slope (rise divided by run of the

r 7 1

Inv Tan137

Inv Tan 1.857 = 61.7

2 2= + = + = =

= = °

3 49 169 218 14 8.

θ

r x y

Inv Tanyx

2 2= +

line), and letter b is the y-axis intercept (the point on the verti-cal axis that the line crosses). For example, Figure 27 is thegraph of the equation y = 2x + 5.

FIGURE 27 ■ Graph of y = 2x + 5.

Example 1: What is the equation of the line in Figure 28?The slope is 12 ÷ 3 = 4 and the y-intercept is -5, so the equa-

tion is y = 4x – 5.Positive slopes in math slant up and to the right; negative

slopes slant down and to the right.

FIGURE 28 ■ Graph requiring an equation.

Example 2: What is the equation of the line in Figure 29?The slope is –6 ÷ 3 = –2 and the y-intercept is 4, so the equa-

tion is y = –2x + 4.

FIGURE 29 ■ Graph requiring an equation.

APPENDIX 5 ■ 13

Page 14: Simbología planos

Oblique Triangles

In trig, lowercase letters usually stand for the lengths of sidesand capital letters stand for angles, as shown in Figure 30.

FIGURE 30 ■ General oblique triangle.

To solve a triangle means to find all missing angles andsides. The Law of Sines can be used to solve a general trianglewhen one side and the opposite angle are known. The Law ofSines is usually expressed as:

It is really three separate equations; each one is a proportion.

Example 1: Find side b in the triangle shown in Figure 31.Angle B can be found quickly knowing that the sum of the

angles of any triangle is 180°.

B = 180 – (80 + 45) = 180 – 125 = 55°Now the Law of Sines can be used:

FIGURE 31 ■ Oblique triangle with unknown side.

aSin A

bSin B

14Sin 45

bSin 55

14.7071

b.8192

.7071b .8192 14

.7071b 11.486b 16.2''

=

°=

°

=

= ( )( )==

aSin A

bSin B

bSin B

cSin C

cSin C

aSin A

=

=

=

a

Sin Ab

Sin Bc

Sin C= =

When an oblique triangle has an angle greater than 90°, caremust be taken when using the Law of Sines to find that angle.The calculator gives only angles less than 90° when INV SIN ispushed. A theorem from trig class must be used: sin θ = sin(180° – θ), which means the calculator answer must be sub-tracted from 180° to get the true answer. The next exampleillustrates this.

Example 2: Find angle A in Figure 32. First, convert to inches:16' – 8'' = 200'' and 14' – 2'' = 170'', then using the version ofthe Law of Sines involving a’s and c’s:

FIGURE 32 ■ Oblique triangle with unknown angle.

Angle A is known to be greater than 90°, so A = 180 –17.7 = 162.3°. This final subtraction from 180 is taken onlywhen finding an angle greater than 90° with the Law ofSines.

Sometimes a side and opposite angle are not available. Inthat case, the triangle may be solved with the Law of Cosines.Again, there are three versions of this law.

a2 = b2 + c2 – 2bc cos Ab2 = a2 + c2 – 2ac cos Bc2 = a2 + b2 – 2ab cos C

aSin A

cSin C

200Sin A

170Sin 15

200Sin A

170.2588

Sin A 200Sin A 51.76

170 Sin A

Sin A .3045A Inv Sin .3045A 17.7

=

=

= ( )( )=

=

=== °

170 2588170

17051 76170

.

.

14 ■ Appendix 5

Page 15: Simbología planos

Example 3: Find side b of the triangle in Figure 33.Use the version of the Law of Cosines containing angle B.

FIGURE 33 ■ Oblique triangle with unknown side.

b 14 2 11 14 cos 130

bbb

b 514.98b 22.7''

2 2

2

2

2

= + − ( )( ) °( )= + − ( )( ) −( )= + +=

==

11

121 196 2 11 14 6428121 196 197 98514 98

2

..

.

Example 4: Find angle A of the triangle in Figure 34.Use the version of the Law of Cosines containing angle A.

FIGURE 34 ■ Oblique triangle with unknown angle.

6 8 2 100 cos A3 cos A3 cos A

3,600 17,569 17,400 cos A13,969 17,400 cos A

17,400 cos A17,400

cos AA Inv cos .8028A 36.6

2 20 100 7 87600 10 000 7 569 17 400600 17 569 17 400

13 96917 400

8028

2= + − ( )( )= + −= −

− = −− = −−−

= −−

=== °

, , , , , , ,

,,

.

APPENDIX 5 ■ 15