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<3D20> Tj 1 0 0 1 324.90 578.88 Tm <2D20> Tj 1 0 0 1 334.62 576.18 Tm <636F7420> Tj 1 0 0 1 350.10 576.18 Tm <30> Tj 0.36 Tc 1 0 0 1 51.12 545.04 Tm <436F2D66756E6374696F6E20> Tj 1 0 0 1 111.42 545.76 Tm <6C64656E74697469657320> Tj 1 0 0 1 158.40 545.94 Tm <3A> Tj 0.72 Tc 1 0 0 1 52.20 527.04 Tm <73696E20> Tj 1 0 0 1 68.58 527.76 Tm <2839302220> Tj 1 0 0 1 91.62 530.28 Tm <2D20> Tj 1 0 0 1 100.26 527.40 Tm <302920> Tj 1 0 0 1 113.94 529.38 Tm <3D20> Tj 1 0 0 1 123.48 527.76 Tm <636F7320> Tj 1 0 0 1 141.66 527.76 Tm <50> Tj 0.36 Tc 1 0 0 1 52.56 508.86 Tm <74616E20> Tj 1 0 0 1 69.84 509.58 Tm <2839302220> Tj 1 0 0 1 92.70 512.10 Tm <2D20> Tj 1 0 0 1 101.52 509.40 Tm <302920> Tj 1 0 0 1 115.20 511.20 Tm <3D20> Tj 1 0 0 1 125.64 509.58 Tm <636F7420> Tj 1 0 0 1 142.02 509.58 Tm <65> Tj 1.08 Tc 1 0 0 1 52.38 490.68 Tm <7365632839302220> Tj 1 0 0 1 92.34 493.74 Tm <2D652920> Tj 1 0 0 1 114.84 493.02 Tm <3D20> Tj 1 0 0 1 124.20 491.22 Tm <63736350> Tj 0.54 Tc /F3 11.0 Tf 1 0 0 1 182.52 527.94 Tm <636F7320> Tj 1 0 0 1 200.70 528.30 Tm <2839306F20> Tj 1 0 0 1 222.84 530.64 Tm <2D20> Tj 1 0 0 1 230.94 527.76 Tm <302920> Tj 1 0 0 1 244.44 529.56 Tm <3A20> Tj 1 0 0 1 253.98 527.94 Tm <73696E20> Tj 1 0 0 1 270.00 527.94 Tm <49> Tj 0.90 Tc 1 0 0 1 182.88 509.76 Tm <636F7420> Tj 1 0 0 1 199.98 510.30 Tm <2839306F20> Tj 1 0 0 1 222.12 512.46 Tm <2D20> Tj 1 0 0 1 230.40 509.58 Tm <302920> Tj 1 0 0 1 243.72 511.38 Tm <3D20> Tj 1 0 0 1 253.62 509.76 Tm <746D29> Tj 0.90 Tc 1 0 0 1 182.70 491.58 Tm <63736320> Tj 1 0 0 1 199.98 491.94 Tm <2839302720> Tj 1 0 0 1 222.30 494.28 Tm <2D20> Tj 1 0 0 1 230.40 491.40 Tm <302920> Tj 1 0 0 1 243.90 493.02 Tm <3D20> Tj 1 0 0 1 253.44 491.58 Tm <73656339> Tj 0.18 Tc 1 0 0 1 438.12 546.48 Tm <503A74746861676F7265616E20> Tj 1 0 0 1 502.38 546.66 Tm <4964656E74697469657320> Tj 1 0 0 1 548.10 546.48 Tm <3A> Tj 1.50 Tc 1 0 0 1 404.10 529.92 Tm <73696E32392B636F7327302D2D49> Tj 1.08 Tc 1 0 0 1 405.36 510.12 Tm <74616E7A20> Tj 1 0 0 1 428.40 510.30 Tm <3020> Tj 1 0 0 1 437.22 510.30 Tm <2B6C3D7365633220> Tj 1 0 0 1 485.10 510.48 Tm <65> Tj 1.08 Tc 1 0 0 1 404.10 490.50 Tm <3120> Tj 1 0 0 1 410.94 490.50 Tm <2B20> Tj 1 0 0 1 421.02 490.50 Tm <636F742720> Tj 1 0 0 1 443.70 490.50 Tm <3020> Tj 1 0 0 1 453.06 492.30 Tm <3D6373632720> Tj 1 0 0 1 485.28 490.68 Tm <30> Tj 0.18 Tc /F3 12.0 Tf 1 0 0 1 50.58 460.44 Tm <53756D20> Tj 1 0 0 1 75.06 458.82 Tm <4C4420> Tj 1 0 0 1 96.48 460.98 Tm <69662E66657220> Tj 1 0 0 1 116.64 461.34 Tm <656E6320> Tj 1 0 0 1 133.38 461.34 Tm <6520> Tj 1 0 0 1 141.30 461.52 Tm <496420> Tj 1 0 0 1 152.64 461.52 Tm <656E74697469657320> Tj 1 0 0 1 189.36 461.52 Tm <3A> Tj 1.26 Tc 1 0 0 1 52.56 446.76 Tm <636F73286120> Tj 1 0 0 1 83.70 447.12 Tm <2B20> Tj 1 0 0 1 93.06 448.38 Tm <70293D636F733675636F737020> Tj 1 0 0 1 173.70 450.36 Tm <2D73696E6173696E4220> Tj 1 0 0 1 297.18 447.66 Tm <636F73286120> Tj 1 0 0 1 327.78 450.54 Tm <2D20> Tj 1 0 0 1 336.42 445.32 Tm <42293D636F7361636F73422B73696E6173696E70> Tj 0.90 Tc 1 0 0 1 52.56 428.94 Tm <73696E286120> Tj 1 0 0 1 82.08 429.30 Tm <2B20> Tj 1 0 0 1 91.62 426.78 Tm <6E20> Tj 1 0 0 1 107.10 431.10 Tm <3D20> Tj 1 0 0 1 116.46 429.66 Tm <73696E6120> Tj 1 0 0 1 142.56 429.84 Tm <636F7320> Tj 1 0 0 1 159.30 431.10 Tm <7020> Tj 1 0 0 1 170.28 430.02 Tm <2B636F736173696E4220> Tj 1 0 0 1 297.54 429.84 Tm <73696E286120> Tj 1 0 0 1 326.52 432.72 Tm <2D20> Tj 1 0 0 1 334.98 427.50 Tm <42293A20> Tj 1 0 0 1 359.46 430.02 Tm <73696E6120> Tj 1 0 0 1 385.02 430.02 Tm <636F7320> Tj 1 0 0 1 401.58 427.68 Tm <4220> Tj 1 0 0 1 412.20 432.90 Tm <2D636F7320> Tj 1 0 0 1 437.94 430.20 Tm <6173696E70> Tj 0.18 Tc 1 0 0 1 121.86 411.84 Tm <746172727120> Tj 1 0 0 1 147.96 412.02 Tm <2B20> Tj 1 0 0 1 158.22 411.84 Tm <74616E20> Tj 1 0 0 1 174.96 409.32 Tm <4220> Tj 1 0 0 1 338.94 409.32 Tm <2C2C5C20> Tj 1 0 0 1 364.50 412.38 Tm <74616E6120> Tj 1 0 0 1 390.78 415.08 Tm <2D20> Tj 1 0 0 1 400.50 412.20 Tm <7461727420> Tj 1 0 0 1 416.88 409.86 Tm <46> Tj 0.72 Tc 1 0 0 1 53.82 403.56 Tm <74616E286120> Tj 1 0 0 1 83.70 404.10 Tm <2B20> Tj 1 0 0 1 92.88 401.40 Tm <422920> Tj 1 0 0 1 108.18 405.90 Tm <3D3B2D20> Tj 1 0 0 1 298.44 404.64 Tm <72616E6C6120> Tj 1 0 0 1 327.96 407.52 Tm <2D20> Tj 1 0 0 1 336.60 405.72 Tm <70293A20> Tj 1 0 0 1 361.80 402.48 Tm <2D> Tj 0.72 Tc 1 0 0 1 101.70 401.58 Tm <2720> Tj 1 0 0 1 118.62 394.92 Tm <6C2D20> Tj 1 0 0 1 133.02 394.92 Tm <74616E20> Tj 1 0 0 1 150.12 394.92 Tm <6174616E20> Tj 1 0 0 1 177.30 396.36 Tm <7020> Tj 1 0 0 1 315.72 402.30 Tm <5C20> Tj 1 0 0 1 336.60 402.12 Tm <2720> Tj 1 0 0 1 345.24 402.48 Tm <2720> Tj 1 0 0 1 361.80 395.46 Tm <4C2B20> Tj 1 0 0 1 376.56 395.46 Tm <74616E20> Tj 1 0 0 1 394.38 395.46 Tm <6174616E20> Tj 1 0 0 1 419.76 396.72 Tm <70> Tj 1.44 Tc /F3 11.0 Tf 1 0 0 1 478.08 277.74 Tm <73696E6120> Tj 1 0 0 1 524.52 277.92 Tm <312D636F7361> Tj 1.08 Tc /F3 12.0 Tf 1 0 0 1 357.66 305.28 Tm <636F733920> Tj 1 0 0 1 387.90 307.26 Tm <3D20> Tj 1 0 0 1 397.98 305.46 Tm <74> Tj 0.00 Tc 1 0 0 1 377.28 296.10 Tm <32> Tj 1.50 Tc 1 0 0 1 358.02 269.82 Tm <315E7320> Tj 1 0 0 1 387.54 271.62 Tm <2D2B> Tj 0.00 Tc 1 0 0 1 376.74 260.46 Tm <32> Tj 0.72 Tc /F3 15.0 Tf 1 0 0 1 408.42 345.06 Tm <4674227222> Tj 1.50 Tc /F3 12.0 Tf 1 0 0 1 402.84 331.38 Tm <5C32> Tj 1.44 Tc /F3 18.0 Tf 1 0 0 1 410.76 309.42 Tm <452B20> Tj 1 0 0 1 430.02 313.20 Tm <2A2222> Tj 0.00 Tc /F3 15.0 Tf 1 0 0 1 405.18 297.36 Tm <5C6C20> Tj 1 0 0 1 432.36 296.28 Tm <32> Tj 1.26 Tc /F3 10.0 Tf 1 0 0 1 410.40 273.78 Tm <5B2D222A22> Tj 1.44 Tc /F3 13.0 Tf 1 0 0 1 404.82 259.92 Tm <5C6C20> Tj 1 0 0 1 414.18 260.64 Tm <312E2E227222> Tj 1.08 Tc /F3 11.0 Tf 1 0 0 1 469.80 260.82 Tm <3120> Tj 1 0 0 1 475.92 260.82 Tm <2B20> Tj 1 0 0 1 485.10 260.82 Tm <636F7361> Tj 0.36 Tc 1 0 0 1 50.58 364.14 Tm <4420> Tj 1 0 0 1 59.94 364.14 Tm <6F75626C652D416E676C6520> Tj 1 0 0 1 121.86 364.86 Tm <6C64656E74697469657320> Tj 1 0 0 1 169.56 364.86 Tm <3A> Tj 0.18 Tc 1 0 0 1 54.18 339.66 Tm <73696E5A637220> Tj 1 0 0 1 86.04 341.64 Tm <2D20> Tj 1 0 0 1 96.12 340.02 Tm <3220> Tj 1 0 0 1 104.22 340.20 Tm <73696E20> Tj 1 0 0 1 119.70 342.36 Tm <712C20> Tj 1 0 0 1 130.32 340.38 Tm <636F7320> Tj 1 0 0 1 147.06 340.38 Tm <642C> Tj 0.72 Tc 1 0 0 1 55.08 304.02 Tm <636F737D613D20> Tj 1 0 0 1 98.10 304.56 Tm <636F733220> Tj 1 0 0 1 121.86 304.74 Tm <6120> Tj 1 0 0 1 132.12 307.62 Tm <2D20> Tj 1 0 0 1 142.92 305.10 Tm <73696E7A20> Tj 1 0 0 1 164.16 304.74 Tm <6420> Tj 1 0 0 1 175.68 306.72 Tm <3D20> Tj 1 0 0 1 186.84 304.92 Tm <5A636F737420> Tj 1 0 0 1 217.08 304.92 Tm <6120> Tj 1 0 0 1 228.60 307.62 Tm <2D20> Tj 1 0 0 1 236.16 304.92 Tm <492D20> Tj 1 0 0 1 252.18 305.10 Tm <312D20> Tj 1 0 0 1 268.38 305.10 Tm <5A73746E7A20> Tj 1 0 0 1 297.18 305.10 Tm <61> Tj 0.36 Tc 1 0 0 1 356.04 231.48 Tm <53756D20> Tj 1 0 0 1 379.80 231.84 Tm <746F20> Tj 1 0 0 1 391.86 231.84 Tm <50726F6475637420> Tj 1 0 0 1 434.34 232.20 Tm <4964656E7469746965733A> Tj 0.90 Tc /F3 20.0 Tf 1 0 0 1 358.02 208.62 Tm <73696E20> Tj 1 0 0 1 373.68 208.62 Tm <6120> Tj 1 0 0 1 383.76 208.98 Tm <2B20> Tj 1 0 0 1 393.12 208.80 Tm <73696E20> Tj 1 0 0 1 409.50 206.10 Tm <4620> Tj 1 0 0 1 421.56 210.60 Tm <3D20> Tj 1 0 0 1 431.46 208.98 Tm <7A2C5E2872792920> Tj 1 0 0 1 497.34 209.16 Tm <2E222C5B2329> Tj 0.72 Tc 1 0 0 1 358.38 172.98 Tm <73696E20> Tj 1 0 0 1 374.04 172.98 Tm <6120> Tj 1 0 0 1 384.12 175.86 Tm <5F20> Tj 1 0 0 1 393.30 173.34 Tm <73696E20> Tj 1 0 0 1 409.68 170.46 Tm <4620> Tj 1 0 0 1 421.74 174.96 Tm <3D20> Tj 1 0 0 1 431.64 173.52 Tm <3220> Tj 1 0 0 1 441.00 173.34 Tm <2A2C287279292C2C225B5429> Tj 0.90 Tc 1 0 0 1 358.74 137.16 Tm <636F7320> Tj 1 0 0 1 375.84 136.98 Tm <6120> Tj 1 0 0 1 386.28 137.34 Tm <2B20> Tj 1 0 0 1 395.46 137.16 Tm <636F7320> Tj 1 0 0 1 413.64 134.64 Tm <4620> Tj 1 0 0 1 425.70 139.14 Tm <3D20> Tj 1 0 0 1 435.24 137.34 Tm <3220> Tj 1 0 0 1 444.78 137.34 Tm <2A2C287279292A7B2329> Tj 0.72 Tc 1 0 0 1 358.92 100.80 Tm <636F7320> Tj 1 0 0 1 376.20 100.62 Tm <6120> Tj 1 0 0 1 386.64 103.68 Tm <5F20> Tj 1 0 0 1 395.64 100.80 Tm <636F7320> Tj 1 0 0 1 413.82 98.28 Tm <4620> Tj 1 0 0 1 425.88 102.60 Tm <3D20> Tj 1 0 0 1 435.60 103.86 Tm <5F322C2A287279292C2D5B2329> Tj 0.18 Tc /F3 11.0 Tf 1 0 0 1 51.30 230.40 Tm <50726F6475637420> Tj 1 0 0 1 94.50 230.94 Tm <746F20> Tj 1 0 0 1 106.92 230.76 Tm <53756D20> Tj 1 0 0 1 130.32 231.12 Tm <6C64656E7469746965733A> Tj 0.72 Tc /F3 12.0 Tf 1 0 0 1 53.64 186.66 Tm <73696E6120> Tj 1 0 0 1 81.00 187.02 Tm <636F7320> Tj 1 0 0 1 98.28 188.46 Tm <7020> Tj 1 0 0 1 109.80 188.82 Tm <2D> Tj 1.44 Tc 1 0 0 1 54.00 153.90 Tm <73696E6173696E503D27> Tj 1.08 Tc /F3 13.0 Tf 1 0 0 1 120.24 195.30 Tm <73696E286120> Tj 1 0 0 1 151.02 195.48 Tm <2B20> Tj 1 0 0 1 160.56 192.96 Tm <42292B73696E20> Tj 1 0 0 1 201.42 195.84 Tm <28612D20> Tj 1 0 0 1 223.92 192.96 Tm <4629> Tj 0.90 Tc /F3 12.0 Tf 1 0 0 1 54.18 120.78 Tm <636F736120> Tj 1 0 0 1 82.98 121.14 Tm <636F7320> Tj 1 0 0 1 100.62 118.62 Tm <4220> Tj 1 0 0 1 111.96 123.12 Tm <2D> Tj 1.50 Tc /F3 9.0 Tf 1 0 0 1 54.00 31.50 Tm <6C6F6632> Tj 0.54 Tc 1 0 0 1 511.38 32.94 Tm <7265766973656420> Tj 1 0 0 1 542.70 33.12 Tm <31316C36313034> Tj ET endstream endobj 4 0 obj 11383 endobj 5 0 obj << /Type /Font /Subtype /Type1 /BaseFont /Helvetica /Encoding /WinAnsiEncoding >> endobj 6 0 obj << /Length 7 0 R >> stream Trigonometric IdentitiesHere is a list of many identities from trigonometry. These identities may be used to verify or establish otheridentities- cos dcot asin d sin (-d) = - sin Icsc(-9) =-ase0 cos (-d) = cos Isec (-d) = sc I tan (-d) - -tan?cot (-d) = - cot 0 Co-function ldentities :sin (90" - 0) = cos Ptan (90" - 0) = cot esec(90" -e) = cscP cos (90o - 0) : sin Icot (90o - 0) = tm)csc (90' - 0) = sec9 P:tthagorean Identities :sin29+cos'0--Itanz 0 +l=sec2 e1 + cot' 0 =csc' 0 Sum LD if.fer enc e Id entities :cos(a + p)=cos6ucosp -sinasinB cos(a - B)=cosacosB+sinasinpsin(a + n = sina cos p +cosasinB sin(a - B): sina cos B -cos asinptarrq + tan B ,,\ tana - tart Ftan(a + B) =;- ranla - p): -' l- tan atan p \ ' ' L+ tan atan p sina 1-cosa cos9 = t21^s -+2 Ft"r"\2 E+ *""\l 2[-"*"\l 1.."r" 1 + cosa D ouble-Angle ldentities :sinZcr - 2 sin q, cos d,cos}a= cos2 a - sinz d = Zcost a - I- 1- Zstnz a Sum to Product Identities: sin a + sin F = z,^(ry) .",[#)sin a _ sin F = 2 *,(ry),,"[T)cos a + cos F = 2 *,(ry)*{#)cos a _ cos F = _2,*(ry),-[#) Product to Sum ldentities: sina cos p -sinasinP=' sin(a + B)+sin (a- F) cosa cos B - lof2 revised 11l6104 endstream endobj 7 0 obj 1157 endobj 8 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE0_TEXTON endstream endobj 9 0 obj << /Length 44 >> stream q 612.18 0 0 792.00 0.00 0.00 cm /Obj2 Do Q endstream endobj 10 0 obj << /Type /Page /MediaBox [ 0 0 612.18 792.00 ] /Parent 11 0 R /Resources << /Font << /F3 5 0 R >> /XObject << /Obj2 2 0 R >> /Subtype 6 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 8 0 R 9 0 R 3 0 R ] >> endobj 13 0 obj << /Type /XObject /Subtype /Image /Width 3401 /Height 4400 /BitsPerComponent 1 /ColorSpace /DeviceGray /Filter /CCITTFaxDecode /DecodeParms << /K -1 /Columns 3401 >> /Length 71790 >> stream >3|x-أ&d'K"LZe"9&4nYd2O-L2)S2i$/aA| ̊Z n`R ɲ| u]yn 3%&a݅ON)#ṫH4Tޘzah9 }Eȧ|*>DX!σgẌ)C>A@@. ~a9ȌĶ\G\ABi uO aB0zi '&@P pgC! 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<70726F647563747320> Tj 1 0 0 1 444.24 544.68 Tm <7573696E6720> Tj 1 0 0 1 473.94 544.86 Tm <74686520> Tj 1 0 0 1 491.94 545.04 Tm <73756D20> Tj 1 0 0 1 513.90 545.04 Tm <746F20> Tj 1 0 0 1 526.32 545.40 Tm <70726F64756374> Tj 0.36 Tc 1 0 0 1 322.56 531.00 Tm <6964656E7469746965732E> Tj 0.36 Tc 1 0 0 1 322.92 488.70 Tm <53696D706C696679696E6720> Tj 1 0 0 1 383.04 488.88 Tm <696E7369646520> Tj 1 0 0 1 415.26 488.70 Tm <6F757220> Tj 1 0 0 1 434.16 489.06 Tm <706172656E74686573657320> Tj 1 0 0 1 494.64 489.60 Tm <676976657320> Tj 1 0 0 1 522.00 489.06 Tm <757320> Tj 1 0 0 1 535.86 488.88 Tm <74686973> Tj 0.36 Tc 1 0 0 1 322.92 475.02 Tm <65787072657373696F6E2E> Tj 0.36 Tc 1 0 0 1 323.28 433.62 Tm <53696E636520> Tj 1 0 0 1 352.98 433.62 Tm <73696E20> Tj 1 0 0 1 370.08 433.98 Tm <282D782920> Tj 1 0 0 1 392.94 436.14 Tm <3A20> Tj 1 0 0 1 402.48 436.14 Tm <2D73696E20> Tj 1 0 0 1 426.42 433.98 Tm <2878292C20> Tj 1 0 0 1 446.40 433.44 Tm <776520> Tj 1 0 0 1 464.04 433.62 Tm <63616E20> Tj 1 0 0 1 483.66 433.62 Tm <7265777269746520> Tj 1 0 0 1 520.38 433.80 Tm <6C696B6520> Tj 1 0 0 1 541.62 433.62 Tm <746869732E> Tj 0.18 Tc 1 0 0 1 323.46 378.18 Tm <416674657220> Tj 1 0 0 1 351.72 378.18 Tm <63616E63656C696E672C20> Tj 1 0 0 1 403.56 378.18 Tm <776520> Tj 1 0 0 1 421.38 378.00 Tm <656E6420> Tj 1 0 0 1 441.90 378.00 Tm <757020> Tj 1 0 0 1 457.20 378.00 Tm <686572652E> Tj 0.18 Tc 1 0 0 1 323.64 337.14 Tm <54686520> Tj 1 0 0 1 345.24 336.96 Tm <6C61737420> Tj 1 0 0 1 365.40 336.96 Tm <7374657020> Tj 1 0 0 1 387.72 336.96 Tm <697320> Tj 1 0 0 1 398.88 336.96 Tm <6120> Tj 1 0 0 1 406.80 336.96 Tm <726174696F20> Tj 1 0 0 1 432.36 336.96 Tm <6964656E746974792E20> Tj 1 0 0 1 476.28 336.96 Tm <5768656E20> Tj 1 0 0 1 507.24 337.32 Tm <776520> Tj 1 0 0 1 524.16 337.14 Tm <6D616E61676520> Tj 1 0 0 1 565.74 336.78 Tm <746F> Tj 0.36 Tc 1 0 0 1 323.46 323.46 Tm <6D616B6520> Tj 1 0 0 1 352.98 323.28 Tm <6F6E6520> Tj 1 0 0 1 373.50 323.28 Tm <7369646520> Tj 1 0 0 1 395.64 323.10 Tm <6F6620> Tj 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1 0 0 1 243.36 142.38 Tm <676976656E20> Tj 1 0 0 1 272.52 141.84 Tm <6F6E20> Tj 1 0 0 1 287.64 141.84 Tm <74686520> Tj 1 0 0 1 302.40 138.96 Tm <666972737420> Tj 1 0 0 1 326.16 142.02 Tm <7061676520> Tj 1 0 0 1 354.24 141.84 Tm <776F756C6420> Tj 1 0 0 1 387.00 141.48 Tm <6F6276696F75736C7920> Tj 1 0 0 1 436.68 141.48 Tm <626520> Tj 1 0 0 1 451.08 141.48 Tm <6120> Tj 1 0 0 1 460.08 141.84 Tm <677265617420> Tj 1 0 0 1 489.60 141.48 Tm <62656E656669742C20> Tj 1 0 0 1 527.76 141.48 Tm <62757420> Tj 1 0 0 1 546.12 141.48 Tm <696620> Tj 1 0 0 1 554.76 141.84 Tm <796F75> Tj 0.00 Tc 1 0 0 1 118.44 127.80 Tm <63616E277420> Tj 1 0 0 1 146.52 127.80 Tm <72656D656D62657220> Tj 1 0 0 1 197.64 127.80 Tm <616C6C20> Tj 1 0 0 1 213.48 127.80 Tm <6F6620> Tj 1 0 0 1 225.36 127.80 Tm <7468656D20> Tj 1 0 0 1 252.36 127.80 Tm <697420> Tj 1 0 0 1 261.36 127.80 Tm <6D617920> Tj 1 0 0 1 285.48 127.80 Tm <7374696C6C20> Tj 1 0 0 1 305.28 127.44 Tm <626520> Tj 1 0 0 1 317.88 127.80 Tm <706F737369626C6520> Tj 1 0 0 1 362.52 127.44 Tm <746F20> Tj 1 0 0 1 374.76 127.44 Tm <646F20> Tj 1 0 0 1 390.24 127.44 Tm <74686520> Tj 1 0 0 1 406.44 127.80 Tm <70726F626C656D2E20> Tj 1 0 0 1 454.32 127.44 Tm <466F7220> Tj 1 0 0 1 476.64 127.44 Tm <696E7374616E63652C20> Tj 1 0 0 1 522.36 127.80 Tm <696E20> Tj 1 0 0 1 534.60 127.44 Tm <746865> Tj 0.00 Tc 1 0 0 1 118.08 113.76 Tm <61626F766520> Tj 1 0 0 1 150.48 113.76 Tm <6578616D706C652C20> Tj 1 0 0 1 196.20 113.76 Tm <74686520> Tj 1 0 0 1 213.84 113.76 Tm <766572696669636174696F6E20> Tj 1 0 0 1 271.44 113.76 Tm <6D617920> Tj 1 0 0 1 294.48 113.40 Tm <626520> Tj 1 0 0 1 308.52 113.40 Tm <6163636F726E706C6B68656420> Tj 1 0 0 1 378.36 113.40 Tm <776974686F757420> Tj 1 0 0 1 417.96 113.40 Tm <7573696E6720> Tj 1 0 0 1 447.84 113.40 Tm <74686520> Tj 1 0 0 1 465.84 113.04 Tm <73756D20> Tj 1 0 0 1 488.52 113.40 Tm <746F20> Tj 1 0 0 1 499.32 113.94 Tm <70726F64756374> Tj 0.00 Tc 1 0 0 1 118.80 99.72 Tm <6964656E74697469657320> Tj 1 0 0 1 166.32 100.44 Tm <28746865736520> Tj 1 0 0 1 197.28 99.72 Tm <61726520> Tj 1 0 0 1 216.36 99.72 Tm <74686520> Tj 1 0 0 1 233.64 99.72 Tm <6F6E657320> Tj 1 0 0 1 258.12 99.72 Tm <4920> Tj 1 0 0 1 265.32 99.72 Tm <6F6A6C656E20> Tj 1 0 0 1 290.16 96.84 Tm <666F72676574292C20> Tj 1 0 0 1 331.56 99.36 Tm <62757420> Tj 1 0 0 1 349.92 99.36 Tm <697420> Tj 1 0 0 1 360.00 99.36 Tm <697320> Tj 1 0 0 1 370.80 99.36 Tm <6D6F726520> Tj 1 0 0 1 398.88 99.36 Tm <74696D652D636F6E73756D696E6720> Tj 1 0 0 1 480.60 99.72 Tm <28747275737420> Tj 1 0 0 1 509.04 99.36 Tm <6D652C20> Tj 1 0 0 1 528.12 99.36 Tm <4920> Tj 1 0 0 1 535.68 99.36 Tm <64696420> Tj 1 0 0 1 554.40 99.00 Tm <692E74> Tj 0.00 Tc 1 0 0 1 118.44 86.04 Tm <7468697320> Tj 1 0 0 1 139.32 85.68 Tm <77617620> Tj 1 0 0 1 162.00 86.04 Tm <74686520> Tj 1 0 0 1 179.28 85.68 Tm <666972737420> Tj 1 0 0 1 201.60 85.68 Tm <74696D642E> Tj 0.90 Tc /F3 10.0 Tf 1 0 0 1 55.62 33.66 Tm <3220> Tj 1 0 0 1 63.90 33.66 Tm <6F663F> Tj 0.54 Tc 1 0 0 1 511.20 33.84 Tm <7265766973656420> Tj 1 0 0 1 543.06 33.66 Tm <4C6C6C36313034> Tj ET endstream endobj 15 0 obj 16920 endobj 16 0 obj << /Length 17 0 R >> stream The following is an example of how to use the given identities to verify another identity. ex: Verify the following: sin 3x - sin x cosx-cos3x = cot Q-)'i. When we are asked to verify (prove) an identity wemust work on each side of the equal signindependently. This means that we cannot use theaddition or multiplication properties of equality forexample, since these involve performing operationson both sides of the equal sign. In this example, woonly work on the left side of the equal sign, but this isnot the only verification possible for this problem.We rewrite our differences in our numerator anddenominator as products using the sum to productidentities.Simplifying inside our parentheses gives us thisexpression.Since sin (-x) : -sin (x), we can rewrite like this. After canceling, we end up here.The last step is a ratio identity. When we manage tomake one side of the equal sign match the other side,we are done with the verification. 2costry)''[?) -"'"(+)'*i+) 2 cos (2")sin(") _@-2cos(2")sink) _@- tt'cot (2x) = cot (2x) Note: Verification of identities is sometimes a slippery endeavor, but here are sorne tips. If you don't lotowhow to start, you might try to make each trigonometric function into a sine or cosine. Sometimes thishelps to make the identities you remember easier to recognize. Also, I Jind thot it is orten easier tobegin on the more complicated side of the equal sign. Notice that in the above example, we startedon the lefi side, which was more complicated than cot(2x) on the right. As you work, keep in mindwhat the goal is. For instance, in the above example, we needed to get an angle of "2x" somewhereon the brt side, since this is the argument of the cotangent given on the right.Note Also: Memorizing the identities given on the first page would obviously be a great benefit, but if youcan't remember all of them it may still be possible to do the problem. For instance, in theabove example, the verification may be accornplkhed without using the sum to productidentities (these are the ones I ojlen forget), but it is more time-consuming (trust me, I did i.tthis wav the first timd. 2 of? revised Lll6104 endstream endobj 17 0 obj 2148 endobj 18 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE0_TEXTON endstream endobj 19 0 obj << /Length 45 >> stream q 612.18 0 0 792.00 0.00 0.00 cm /Obj13 Do Q endstream endobj 20 0 obj << /Type /Page /MediaBox [ 0 0 612.18 792.00 ] /Parent 11 0 R /Resources << /Font << /F3 5 0 R >> /XObject << /Obj13 13 0 R >> /Subtype 16 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 18 0 R 19 0 R 14 0 R ] >> endobj 11 0 obj << /Kids [ 10 0 R 20 0 R ] /Count 2 /Type /Pages >> endobj 12 0 obj << /Pages 11 0 R /Type /Catalog >> endobj xref 0 21 0000000000 65535 f 0000000015 00000 n 0000000110 00000 n 0000057652 00000 n 0000069089 00000 n 0000069111 00000 n 0000069214 00000 n 0000070424 00000 n 0000070445 00000 n 0000070523 00000 n 0000070618 00000 n 0000162531 00000 n 0000162604 00000 n 0000070867 00000 n 0000142876 00000 n 0000159852 00000 n 0000159875 00000 n 0000162078 00000 n 0000162100 00000 n 0000162179 00000 n 0000162276 00000 n trailer << /Size 21 /Info 1 0 R /Root 12 0 R /ID [<692984c71f7027e9266a20a6d6a48379><692984c71f7027e9266a20a6d6a48379>] >> startxref 162659 %%EOF