Additional Mathematics Notes

Additional Mathematics Notes Functions and Graphs 2𝑥#+2 𝑎>0 Turning point = Minimum point −2𝑥#+2 𝑎<0 Turning point = Maximum point Line of symmetry always passes through the turning point (𝑥-coordinate of turning point/mid-value of 𝑥-intercepts) y-intercept, x=0 𝑦=𝑎𝑥#+𝑏𝑥+𝑐 𝑦=𝑎(0)#+𝑏(0)+𝑐 x-intercept, y=0 0=𝑎𝑥#−𝑏𝑥+𝑐 Completing the Square 𝑎𝑥#+𝑏𝑥+𝑐 𝑎(𝑥#+𝑏𝑎𝑥+𝑐𝑎) 𝑎0(𝑥+𝑏2𝑎)#−1𝑏2𝑎2#+𝑐𝑎3 𝑎(𝑥+ℎ)#+𝑘 𝑎(𝑥−ℎ)#+𝑘 Coefficient of 𝑥# must be +1 𝑎𝑥#−𝑏𝑥+𝑐 𝑎(𝑥#−𝑏𝑎𝑥+𝑐𝑎) 𝑎0(𝑥−𝑏2𝑎)#−1𝑏2𝑎2#+𝑐𝑎3 Conditions for Curve to Lie Completely Above or Below x-axis Above Coefficient of 𝑥#>0 Minimum value is positive 𝑏#−4𝑎𝑐<0 ∴Since the coefficient of 𝑥#=1>0 and the minimum value of 𝑦=4, which is positive, the curve lies completely above the x-axis. Below Coefficient of 𝑥#<0 Minimum value is negative ∴Since the coefficient of 𝑥#=−1<0 and the maximum value of 𝑦=−4, which is negative, the curve lies completely below the x-axis. Quadratic Formula (Given) 𝑎𝑥#+𝑏𝑥+𝑐 𝑥=−𝑏±√𝑏#−4𝑎𝑐2𝑎 Nature of Roots of Quadratic Equations 𝑏#−4𝑎𝑐≥0 Real roots 𝑏#−4𝑎𝑐>0 2 real and distinct roots 𝑏#−4𝑎𝑐<0 Non-real roots 𝑏#−4𝑎𝑐=0 2 real and equal roots Intersection Between Line and Curve 𝑏#−4𝑎𝑐>0 Line cuts curve at 2 distinct points 𝑏#−4𝑎𝑐=0 Line cuts curve at 1 point (Line is tangent to curve) 𝑏#−4𝑎𝑐<0 Line does not intersect curve Finding Range of Values E.g. (𝑝−6)(𝑝+2)>0 Draw number line and curve Laws of Surds √𝑎𝑏 √𝑎×√𝑏 ?𝑎𝑏 √𝑎√𝑏 √𝑎×√𝑎, (√𝑎)# 𝑎 𝑝√𝑎+𝑞√𝑎 (𝑝+𝑞)√𝑎 𝑝√𝑎−𝑞√𝑎 (𝑝−𝑞)√𝑎 𝑝#+(𝑞√𝑎)# 𝑝#+𝑞#𝑎 1√2+3 1√2+3×√2−3√2−3 √2−32−9 −√2−37 Equality of Surds 𝑎+𝑏√𝑛 5+4√𝑛 𝑎=5 𝑏=4

Polynomials Degree of 𝑃(𝑥)×𝑄(𝑥) Degree of 𝑃(𝑥) + Degree of 𝑄(𝑥) Division Algorithm: 𝑃(𝑥)=𝐷(𝑥)×𝑄(𝑥)+𝑅(𝑥) 𝑃(𝑥) = Dividend 𝐷(𝑥) = Divisor 𝑄(𝑥) = Quotient 𝑅(𝑥) = Remainder Remainder and Factor Theorem 𝑓(𝑥)÷(𝑎𝑥+𝑏) 𝑅𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟=𝑓(−𝑏𝑎) Factor Theorem, Remainder=0 Cubic Expressions and Equations 𝑎𝑥R+𝑏𝑥#+𝑐𝑥+𝑑 (𝑎𝑥+𝑏)(ℎ𝑥+𝑘)(𝑝𝑥+𝑞) (𝑎𝑥+𝑏)(𝑝𝑥#+𝑞𝑥+𝑟) Step 1 Factorise 𝑓(𝑥) using Factor Theorem [Mode-3-4] Step 2 Use Synthetic Division/ Long Division/ Comparing Coefficients to factorise 𝑓(𝑥) completely Synthetic Division 𝑓(𝑥)=2𝑥R−11𝑥#−7𝑥+6 -1 2 -11 -7 6 -2 13 -6 2 -13 6 0 (𝑥+1) 𝑓(𝑥)=(𝑥+1)(2𝑥#−13𝑥+6) Cubic Identities Sum of cubes 𝑎R+𝑏R=(𝑎+𝑏)(𝑎#−𝑎𝑏+𝑏#) Difference of cubes 𝑎R−𝑏R=(𝑎−𝑏)(𝑎#+𝑎𝑏+𝑏#) Partial Fractions Step 1: Fraction must be a proper fraction (Degree of numerator < Degree of denominator) Step 2: Factorise denominator Step 3: Express proper algebraic fraction in partial fractions Case Denominator Proper Fraction Partial Fraction 1 Distinct linear factors 𝑝𝑥+𝑞(𝑎𝑥+𝑏)(𝑐𝑥+𝑑) 𝐴𝑎𝑥+𝑏+𝐵𝑐𝑥+𝑑 2 Repeated linear factors 𝑝𝑥+𝑞(𝑎𝑥+𝑏)# 𝐴𝑎𝑥+𝑏+𝐵(𝑎𝑥+𝑏)# 3 Quadratic factor (cannot be factorised) 𝑝𝑥#+𝑞𝑥+𝑟(𝑎𝑥+𝑏)(𝑥#+𝑐#) 𝐴𝑎𝑥+𝑏+𝐵𝑥+𝐶𝑥#+𝑐# Step 4: Solve for unknown constant Binomial Theorem (Given) Binomial expansion of (𝑎+𝑏)V W𝑛0X𝑎V+W𝑛1X𝑎VYZ𝑏+W𝑛2X𝑎VY#𝑏#+⋯+W𝑛𝑟X𝑎VY\𝑏\+W𝑛𝑛−1X𝑎𝑏VYZ+(𝑛𝑛)𝑏V General Term 𝑇\^Z=W𝑛𝑟X𝑎VY\𝑏\ Law of Indices 𝑎_=1 𝑎`×𝑎V=𝑎`^V 𝑎YV=1𝑎V 𝑎`÷𝑎V=𝑎`YV 𝑎`V=√𝑎`a (𝑎`)V=𝑎`V (𝑎𝑏)V=𝑎V𝑏V (𝑎𝑏)V=𝑎V×𝑏V 𝑦=𝑎b, 𝑦>0, 𝑎>0 Converting Exponential to Logarithmic 𝑦=𝑎b 𝑥=logf𝑦

Law of Logarithms logf1=0 logf𝑥+logf𝑦=logf𝑥𝑦 logf𝑎=1 logf𝑥−logf𝑦=logf𝑥𝑦 𝑎ghijk=𝑦, 𝑦>0 logf𝑥\=𝑟logf𝑥 Change of Base logf𝑏=logl𝑏logl𝑎 Exponential Functions and Graphs 𝑦=𝑏b, where b>1 𝑦=𝑏b, where 0<𝑏<1 Logarithmic Functions and Graphs 𝑦=logm𝑥, where 𝑏>0 𝑦=logm𝑥, where 0<𝑏<1

Coordinate Geometry Midpoint of Line (𝑥Z+𝑥#2,𝑦Z+𝑦#2) Length of Line o(𝑥#−𝑥Z)#+(𝑦#−𝑦Z)# Gradient of line 𝑦#−𝑦Z𝑥#−𝑥Z tan𝜃 Gradient of Perpendicular Lines 𝑚Z×𝑚#=−1 Equation of Straight Line 𝑦=𝑚𝑥+𝑐 𝑦−𝑦Z=𝑚(𝑥−𝑥Z) Area of Polygon 12t𝑥Z𝑦Z 𝑥#𝑦# 𝑥R𝑦R 𝑥u𝑦u 𝑥Z𝑦Zt (anti-clockwise) Equation of Circle (𝑥−𝑎)#+(𝑦−𝑏)#=𝑟# 𝑥#+𝑦#+2𝑔𝑥+2𝑓𝑦+𝑐=0, where 𝑔#+𝑓#−𝑐>0, Centre (−𝑔,−𝑓), Radius 𝑟=o𝑔#+𝑓#−𝑐 Linear Law 𝑌=𝑚𝑋+𝑐 𝑌 and 𝑋 only contains 𝑥 and 𝑦 𝑚 and 𝑐 only contains constants Sketching Pointers Radian Measure 𝜋 𝑟𝑎𝑑=180° Special Angles 0° (0) 30° W𝜋6X 45° W𝜋4X 60° W𝜋3X 90° W𝜋2X sin𝜃 √02=0 √12=12 √22 √32 √42=1 cos𝜃 1 √32 √22 12 0 tan𝜃 0 √33 1 √3 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 sin𝜃→ √#, 0 𝑡𝑜 4 cos𝜃→sin𝜃 flipped opposite tan𝜃→ ƒ„…†‡hƒ† General Angles and Basic Angles Basic angle, 𝛼→ General angle, 𝜃 1st Quadrant 𝜃=𝛼 𝜃=𝛼 2nd Quadrant 𝜃=180°−𝛼 𝜃=𝜋−𝛼 3rd Quadrant 𝜃=180°+𝛼 𝜃=𝜋+𝛼 4th Quadrant 𝜃=360°−𝛼 𝜃=2𝜋−𝛼 Cosecant, Secant and Cotangent cosec𝜃→1sin𝜃 sec𝜃→1cos𝜃 cot𝜃→1tan𝜃→cos𝜃sin𝜃

Maximum Minimum Amplitude 𝑦=cos𝑥 Period Trigonometric Ratios of General Angles sin𝜃=k\, cosec𝜃=Zƒ„…†=\k cos𝜃=b\, sec𝜃=Z‡hƒ†=\b tan𝜃=kb, cot𝜃=ZŠ‹…†=bk 𝑥=𝑥-coordinate of point 𝑃 𝑦=𝑦-coordinate of point 𝑃 𝑟=o𝑥#+𝑦# tan𝜃=ƒ„…†‡hƒ†, where cos𝜃≠0 Signs of Trigonometric Ratios & Trigonometric Ratios of Related Angles (ASTC) 2nd Quadrant (S) 1st Quadrant (A) 𝑥<0 𝑦>0 𝑟>0 𝐬𝐢𝐧𝜽→ positive cos𝜃→ negative tan𝜃→ negative sin𝜃=sin𝛼 cos𝜃=−cos𝛼 tan𝜃=−tan𝛼 𝑥>0 𝑦>0 𝑟>0 sin𝜃→ positive cos𝜃→ positive tan𝜃→ positive sin𝜃=sin𝛼 cos𝜃=cos𝛼 tan𝜃=tan𝛼 3rd Quadrant (T) 4th Quadrant (C) 𝑥<0 𝑦<0 𝑟>0 sin𝜃→ negative cos𝜃→ negative 𝐭𝐚𝐧𝜽→ positive sin𝜃=−sin𝛼 cos𝜃=−cos𝛼 tan𝜃=tan𝛼 𝑥>0 𝑦<0 𝑟>0 sin𝜃→ negative 𝐜𝐨𝐬𝜽→ positive tan𝜃→ negative sin𝜃=−sin𝛼 cos𝜃=cos𝛼 tan𝜃=−tan𝛼 Relationship between Trigonometric Ratio of General Angle 𝜽 and its Negative Angle −𝜽 sin(−𝜃)=−sin𝜃 cos(−𝜃)=cos𝜃 tan(−𝜃)=−tan𝜃 Trigonometric Ratios of Complementary Angles sin(90°−𝜃)=cos𝜃 cos(90°−𝜃)=sin𝜃 tan(90°−𝜃)=1tan𝜃 ↓ cot𝜃→1tan𝜃→cos𝜃sin𝜃 Properties of Sine, Cosine and Tangent Graphs 𝑦=tan𝑥 Special Values of 1 tan𝑥=1 When 𝑥=45°,225° 𝑥=𝜋4,5𝜋4 Special Values of -1 tan𝑥=−1 When 𝑥=135°,315° 𝑥=3𝜋4,7𝜋4 Amplitude Period Maximum Minimum 𝑦=sin𝑥 Period 𝑦=tan𝑥

Properties of Sine, Cosine and Tangent Graphs (Basic Graph) 𝑦=sin𝑥 𝑦=cos𝑥 𝑦=tan𝑥 Zero Value sin𝑥=0, When 𝑥=0°,180°,360° 𝑥=0,𝜋,2𝜋 cos𝑥=0 When 𝑥=90°,270°,•#,R•# tan𝑥=0 When 𝑥=0°,180°,360° 𝑥=0,𝜋,2𝜋 Maximum Value sin𝑥=1, When 𝑥=90°,•# cos𝑥=1 When 𝑥=0°,360°,0,2𝜋 Can be any real number, No range or max/min value Minimum Value sin𝑥=−1 When 𝑥=270°,R•# cos𝑥=−1 When 𝑥=180°,2𝜋 Range −1≤sin𝑥≤1 −1≤cos𝑥≤1 Axis of Curve 𝑦=0 𝑦=0 Period 360°, 2𝜋 360°, 2𝜋 180°, 𝜋 Amplitude 1 1 Symmetry Rotational symmetry of order 2 about the origin sin(−𝑥)=−sin𝑥 Symmetrical about 𝑦-axis cos(−𝑥)=cos𝑥 Rotational symmetry of order 2 about the origin tan(−𝑥)=−tan𝑥 𝒚=𝒂𝐬𝐢𝐧𝒃𝒙+𝒄, 𝒚=𝒂𝐜𝐨𝐬𝒃𝒙+𝒄 and 𝒚=𝒂𝐭𝐚𝐧𝒃𝒙+𝒄 𝑦=𝑎sin𝑏𝑥+𝑐, 𝑦=𝑎cos𝑏𝑥+𝑐 𝑦=𝑎tan𝑏𝑥+𝑐 Maximum Value 𝑎+𝑐 Minimum Value −𝑎+𝑐 Amplitude 𝑎=𝑚𝑎𝑥−𝑚𝑖𝑛2 Period Rœ_°m or #•m Z_°m or •m Interval 𝑃𝑒𝑟𝑖𝑜𝑑4 Vertical Shift 𝑐=𝑚𝑎𝑥+𝑚𝑖𝑛2 𝑐 Sketching of Graphs Step 1: Find the period Step 2: Find the interval Step 3: Recall 5 critical points for the correct trigonometric function Graph 5 Critical Points Sine 0 1 0 -1 0 Cosine 1 0 -1 0 1 Tangent 0 1 Undefined -1 0 Step 4: Multiply 5 critical points by 𝑎 Step 5: Add 𝑐 to the values in Step 4 Step 6: Use values in Step 5 to mark and sketch out the graph Principle Values Degree Radian Value of 𝑥 sinYZ𝑥 −90°≤sinYZ𝑥≤90° −𝜋2≤sinYZ𝑥≤𝜋2 Where 𝑥=−1≤𝑥≤1 cosYZ𝑥 0°≤cosYZ𝑥≤180° 0≤cosYZ𝑥≤𝜋 tanYZ𝑥 −90°≤tanYZ𝑥≤90° −𝜋2≤tanYZ𝑥≤𝜋2 Where 𝑥 is any real value Basic Trigonometric Identities (Given) sin#𝜃+cos#𝜃=1 sec#𝜃=1+tan#𝜃 cosec#𝜃=1+cot#𝜃

For 0°≤𝑥≤360°, 0 1 −1 sin𝑥 sin𝑥=0, 𝑥=0°,180°,360° sin𝑥=1, 𝑥=90° sin𝑥=−1 𝑥=270° cos𝑥 cos𝑥=0, 𝑥=90°,270° cos𝑥=1 𝑥=0°,360° cos𝑥=−1, 𝑥=180° tan𝑥 tan𝑥=0, 𝑥=0°,180° tan𝑥=1, 𝑥=45°,225° tan𝑥=−1, 𝑥=135°,315° For 0≤𝑥≤2𝜋 0 1 −1 sin𝑥 sin𝑥=0, 𝑥=0,𝜋,2𝜋 sin𝑥=1, 𝑥=𝜋2 sin𝑥=−1 𝑥=3𝜋2 cos𝑥 cos𝑥=0, 𝑥=𝜋2,3𝜋2 cos𝑥=1 𝑥=0,2𝜋 cos𝑥=−1, 𝑥=𝜋 tan𝑥 tan𝑥=0, 𝑥=0,𝜋 tan𝑥=1, 𝑥=𝜋4,5𝜋4 tan𝑥=−1, 𝑥=3𝜋4,7𝜋4 Solving Trigonometry Equations Questions Example Solve the equation sin𝑥=0.5, where 0°≤𝑥≤360°. sin𝑥=0.5 0°≤𝑥≤360° 𝑥 lies in 1st or 2nd quadrant Basic angle, sin𝛼=0.5 𝛼=sinYZ0.5 𝛼=30° 𝑥=30°, 180°−30° =30°, 150°