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1、Maths Extension 1 Trigonometryhttp:/ 1Trigonometry Trigonometric Ratios Exact Values & Triangles Trigonometric Identities ASTC Rule Trigonometric Graphs Sine & Cosine Rules Area of a Triangle Trigonometric Equations Sums and Differences of angles Double Angles Triple Angles Half Angles T formula Sub
2、sidiary Angle formula General Solutions of Trigonometric Equations Radians Arcs, Sectors, Segments Trigonometric Limits Differentiation of Trigonometric Functions Integration of Trigonometric Functions Integration of sin2x and cos2x INVERSE TRIGNOMETRY Inverse Sin Graph, Domain, Range, Properties In
3、verse Cos Graph, Domain, Range, Properties Inverse Tan Graph, Domain, Range, Properties Differentiation of Inverse Trigonometric Functions Integration of Inverse Trigonometric FunctionsMaths Extension 1 Trigonometryhttp:/ 2Trigonometric RatiosSine sin= hypotenusiCosine cos = adjcTangent tan= jentsiC
4、osecant cosec = sin1= opsihyuSecant sec= co= adjcentCotangent cot = tan1= opsisin= 90coscos = intan = tcosec = secsec= 90ocot = tan60 seconds = 1 minute 60 = 160 minutes = 1 degree 60 = 1cosintasincothypotenuse hypotenuseoppositeadjacentadjacentoppositeMaths Extension 1 Trigonometryhttp:/ 3Exact Val
5、ues & Triangles0 30 60 45 90 180sin0 2123211 0co1 31 0 1ta0 1 1 0ecs 2 32 1 ec1 32 2 1ot 311 0 Trigonometric Identities 22cossin= 1= 2sini= co2cot1= cosec2= cosec2 1 1 = cosec2 2cottan2= sec= 121 = 2tan112453123060Maths Extension 1 Trigonometryhttp:/ 4ASTC RuleFirst Quadrant: All positivesinsin+coco
6、+tata+Second Quadrant: Sine positive180sinsin+cocota taThird Quadrant: Tangent positive180sin sincocotata+Fourth Quadrant: Cosine positive360sin sincoco+ta ta36090180270S AT C1st Quadrant4th Quadrant2nd Quadrant3rd QuadrantMaths Extension 1 Trigonometryhttp:/ 5Trigonometric GraphsSine & Cosine Rules
7、Sine Rule:ORCcBbAasinisincCbBaAsinisinCosine Rule: Abcaos22ABC abcAabcMaths Extension 1 Trigonometryhttp:/ 6Area of a TriangleCbAsin21 C is the angle & are the two adjacent sidesabCbaMaths Extension 1 Trigonometryhttp:/ 7Trigonometric Equations Check the domain eg. 360 Check degrees ( ) or radians (
8、 )0 2 If double angle, go 2 revolutions If triple angle, go 3 revolutions, etc If half angles, go half or one revolution (safe side)Example 1Solve sin = for 2360sin= 21= 30, 150Example 2Solve cos 2 = for 21360cos= = 60, 300, 420, 660= 30, 150, 210, 330Example 3Solve tan = 1 for 2 360tan 2= 1= 45, 22
9、5= 90Example 4 0cos2sin= 01i= 0cos= 0 sin= 21= 90, 270= 210, 330Example 5 2cosin3in1= 2Maths Extension 1 Trigonometryhttp:/ 81sin3i2= 0= 0si= 21sin= 1= 210, 330= 270Maths Extension 1 Trigonometryhttp:/ 9Sums and Differences of anglessin= sincosin= co= ics= sstan= tan1t= Double Angles2sin= cosin2co=
10、2i= s1= co2tan= 2tan12si= cos21co= Triple Angles3sin= 3sin4ico= co3ta= 2ta1Half Anglessin= 2cosinco= i= 2s1Maths Extension 1 Trigonometryhttp:/ 10= 1cos2tan= 2tanMaths Extension 1 Trigonometryhttp:/ 11Deriving the Triple Angles3sin= 2sin= sin2co= si1i 2= 3iis= 2in= 33sinii= s4i_Normal double angle_E
11、xpand double angle_Multiply_Change _1cossin22Simplify_3cos= 2cos= sin= sico2c1= is3= o2= 3csc= s433tan= 2tan= t1= 2tan1t2= 23tan1t= 3Maths Extension 1 Trigonometryhttp:/ 12T FormulaeLet t = tan 2sin= 21tco= tan= 21tsin= 2cosi= in= 22cosins= 2tan1= Using half angles_Divide by “1”1cossin22Divide top a
12、nd bottom by 2cos cancel; becomes tancoscosincos= 22sinco= 22i= 22cosins= 2tan1= ttan= si= 21tt= 2tMaths Extension 1 Trigonometryhttp:/ 13Subsidiary Angle Formula xbacossin= )sinco(sinxxR= = Rs2a= x2csb= xinb= in1co22= 2R2baR abtnxbacossin= C )sin(xR= C i= C cos= C )s(Example 1Find x. 1cosin3xR = 2t
13、an= 31= 4= 2 = 30)30sin(2xx= 1= 2= 30, 150= 60, 180Maths Extension 1 Trigonometryhttp:/ 14General Solutions of Trigonometric EquationssiniThen n)1(coThen 2tantThen nRadiansc= 1801 = 180cArcs, Sectors, SegmentsArc Lengthl= rArea of SectorA = 21r l r rMaths Extension 1 Trigonometryhttp:/ 15Area of SegmentA = sin21r rSegmentMaths Extension 1 Trigonometryhttp:/ 16Trigonometric Limitsxsinlm0= xtanlim0= xxcoslim0= 1Differentiation of Trigonometric Functionsxdsin= xcos)(sifx= )(ff)sin(bad= )cos(baxxcos= sin)(sfdx= )(ixff)cos(ba= )sin(baxdtn= x2sec)(tafx= )()(2ff)tn(bd= )(sec2baxxsec=
