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ACADEMIC SKILLS CENTRE (ASC) Formula for Success a Mathematics Resource Peterborough • Oshawa Contents Section 1: Formulas and Quick Reference Guide 1. Formulas From Geometry 2. Interval Notation 3. Coordinate Geometry 4. Steps in completing the Square 5. The Graphs of the Eight Basic Functions 6. Radian Measure 7. Special Angles 8. The Unit Circle 9. Inverse Trig Functions 10. Trigonometric Identities 11. Sine Law 12. Cosine Law 13. Graphs of Trig Functions Section 3: Calculus/Algebra Prep 1 3 4 5 6 8 9 10 11 12 13 14 15 Section 2: Mathematics Review Modules 1. Order of Operations and Substitution. 2. Fractions, Decimals and Percent 3. Exponents, Roots and Factorials For information on this or any of our services, contact the Academic Skills Centre, Trent University, Peterborough, Ontario K9J 7B8 16 20 27 1. i) ii) iii) Algebra Terminology 32 Expanding and Factoring 33 Factor Theorem and Synthetic Division 38 2. i) ii) iii) iv) v) 40 41 42 43 Equations Definitions Linear Equations X and Y intercepts Solving Quadratics Using Factoring Solving Quadratics Using Quadratic Formula vi) Solving Absolute Value Equations vii) Solving Polynomial Equations 3. Exponents i) Exponents ii) The Exponential Function and its Inverse the Logarithmic Function iii) The Common Logarithm iv) The Natural Logarithm v) Laws of Logarithms vi) Solving Equations with Exponents or Logarithms vii) Expoential Growth and Decay 4. i) ii) 5. i) ii) iii) 44 46 47 49 51 53 53 54 56 58 Functions and Transformations Definition of a function Function Notation 60 61 Inequalities Inequalities Definition Linear Inequalities Polynomial Inequalities 62 62 63 6. Trigonometric Functions i) Trig Definitons 66 ii) CAST Rule 68 iii) Radian Measure 70 iv) Finding Exact Values of Trig Ratios 71 v) Using a calculator to find Trig Ratios 74 vi) Trig Equations 76 vii) Trig Identities 77 viii) Graphs of y = sinx and y = cosx 80 Phone (705) 748-1720 Fax (705) 748-1830 e-mail: [email protected] www.trentu.ca/academicskills The Academic Skills Centre would like to thank the following for their contributions to this publication: Ruth Brandow, Ellen Dempsey, Marj Tellis, Lisa Davies. Section 1 Formulas and Quick Reference Guide 1. Formulas from Geometry Pythagoream Theorem If a triangle has sides a, b, and c (c is the hypotenuse), then c2 = a2 + b2 Triangle with Base and Height Given The area of a triangle whose base is b and altitude is h (the perpendicular distance to the base) is A = ½ b x h Triangle with Base and Height Unknown The area of a triangle with angles A, B, and C and sides opposite a, b, and c, respectively: Area = ½ ab x sinC Area of a Parallelogram The area of a parallelogram with base b and altitude h is Area = bh Area of a Trapezoid The area of a trapezoid whose parallel sides are a and b and altitude is h is Area = ½(a + b)h 1 Circle Formulae For a circle whose radius is r and diameter is d (d = 2r): Circumference : C = 2pr= pd Area: A = pr2 If s is the length of an arc subtended by a central angle of q radians, then s = rq 3 Dimensional Solids Sphere A Sphere with radius r has: Surface Area = 4pr2 Volume = 4/3 pr3 Right Circular Cylinder A right circular cylinder where r is the radius of the base and h is the altitude has: Surface area = 2prh Volume = pr2h Right Circular Cone A right circular cone where r is the radius of the base and h is the altitude has: Surface area = p r2+ h2 Volume = 1/3 pr2h 2 2. Interval Notation { ≤ ≤ } { ≤ < } { < ≤ } ∞ ∞ ∞ ≥ ∞ ≥ ∞ ∞ 3 3. Co-ordinate Geometry = = − − − + − + + + − 4 4. Steps in Completing the Square + + = + + + + = + + − + − − = + + + + = + + + + = + + − + = + + − + = − + = − + − − − + = − − = − − − − + = − − + − − + = − − + = − − − 5 5. The Graphs of the Eight Basic Functions 6 7 6. Trigonometry - Radian Measure πππ π ππ π π π π = π = π π 8 7. Special Angles Finding The Exact Value of a Trigonometric Ratio 9 8. The Unit Circle 10 9. Inverse Trigonometric Functions (a.k.a. arcsin, arccos and arctan) π π − − ≤ ≤ − π ≤≤ π [ π ] − ≤ ≤ ≤ ≤ π π π π π − − 11 10. Trigonometric Identities = + − − + − 12 11. Sine Law 13 12. Cosine Law + − 14 13. Graphs of Trigonometric Functions 15 Section 2 Mathematics Review Modules 1. Order of Operations and Substitution Order of Operations + − − ÷ + 10 12÷÷+3+ 2 ++2(4)−− == 10++8− − 12 ÷ ÷+3+ 2 == 10++8− −+4+ 2 == 18+ −+4+ 2 == 14+ + 2 == 16 16 + + = = / + = / 17 Substituting into Formulae + + + = = = For Practice Problems on this section, please visit: www.trentu.ca/academicskills/online_math.php 18 19 2. Fractions, Decimals, and Percents Includes Rounding and Significant Digits Fractions • • • ÷= = • • • 20 = = × = × = = ÷ = = + + = ÷ = × = = 21 + + = + + = + = + + = + = = 22 Percents 23 Decimals ÷= ÷ ÷ 24 Rounding and Significant Digits • • 25 • • • • • • For Practice Problems on this section, please visit: www.trentu.ca/academicskills/online_math.php 26 3. Powers, Roots and Factorials Powers 27 28 = + = = = Roots + ≠ + = = 29 Laws of Exponents ⇒ ⋅ = ÷ = − ⇒ ÷ =− ( ) = = ⇒( ) = = = ⇒( −) = ⇒− =− ⇒− = = = ⇒ = = = ⇒ = = = − − ⇒ = = = 30 − ⇒ = ⋅ = − = = ⇒ = = Factorials / / / / / = / / / / / = = = ⋅ = = For Practice Problems on this section, please visit: www.trentu.ca/academicskills/online_math.php 31 Section 3 Calculus/Algebra Prep 1. Essential Algebra i) Terminology − + − − + − 32 − + ≠ − + + + = ii) Expanding and Factoring − + − − + − 33 + − − + ( − + ) 34 − − = − + = + − + + + + + − − ∴ − − = − + + − ∴ + − = + − 35 + + + + ≠ ≠ ≠ ≠ + + + + + −− + ∴ ∴ + −− == − −− + + +− + + ( + ) − + ( − ) 36 ( ) + − + = + − + − = − + + − = − + + + = + − + 37 iii) Factor Theorem (polynomial of degree three or greater) + − + ÷ − − = ∴ = ∴ + + + − + = − + + + − + − ∴ − − − + − − = 38 − −−− − − + + ∴ ∴ − −− − − −÷ ÷ + + − − − − − + + − − + ++ ++ + − − + + + − − + + + ∴ ∴ + +−−++ For Practice Problems on this section, please visit: www.trentu.ca/academicskills/online_math.php 39 2. Equations of Straight Lines i) Definitions − − − + − 40 Equations ii) Linear Equations ∴ − + ∴ − + − = () − = − = + − − = () − + 41 iii) X and Y intercepts ∴ ≠ ∴ 42 iv) Quadratic Equations Using Factoring − + = ∴ ∴ = − + = ∴ ∴ + − += + − + − − + = + − + − − = ∴ ∴ − 43 v) Solving Quadratics Using Quadratic Formula = − ± − + + = + + − = − ( + ) − = − ( + ) = − ( + ) = − + = ± − = − ± − − − = = 44 ± − − ∴ = ± ∴ = ± ∴ = ± ∴ = ± − = − = ± − ∴ = ± − − ∴ = ± − − = − = − − = 45 vi) Absolute Value Equations = = − = ⇒ ≥ = < = − + = ⇒ ∴ ⇒ ∴ − = − ⇒ ∴ = ≥ ⇒ ⇒ ∴ = < 46 − vii) Solving Polynomial Equations (With Degree Three or More) + + = + + = + + = + = + − + = ± = − = − = ± − = ± 47 − + = = − + ∴ = − + − ∴ = − + − ∴ − + − = ∴ = = − = − + = − − = − + − + = ∴ ∴ = ± = ± ∴ ∴ ∴ ∴ ∴ − + + = ∴ − − + − + − − + − = ∴ − − + − = = − [ ] ∴ − − + − − = ∴ − + − = For Practice Problems on this section, please visit: www.trentu.ca/academicskills/online_math.php 48 3. Exponents i) Laws of Exponents ≠ = + ⇒ ⋅ = ÷ = − ⇒ ÷ = − = ⇒ = ⋅ = = () = ⇒ = × = = − ⇒ ( ) = = ⇒ − = ⇒ − = − − = = = ⇒ − = = ⇒ = = ⇒ = = = − ⇒ = − = = − = − 49 − = − − − − = − − − − = = − = − − − = − = − = − = − = ( ) = = 6. ( − ) − ( ) − − = − − = ( ) 50 ( ) − − = ( ) =( ) − = − ( − − − = = = ) ) − ( ( ) = = − − − ( ) − − − = = ii) The Exponential Function and Its Inverse The Logarithmic Function = = = = = = ⇒ = = = = = ⇒ = 51 = = = = = = = = = = = = = = = = − () − = = = = = − = = − = = 52 iii) The Common Logarithm = = − = = iv) The Natural Logarithm (ln and e) = 53 v) Laws of Logarithms = + ⇒ = + = − = = = ⇒ ( ) = − ⇒ = ⇒ = ⇒ = + [ = ] = − = ( ) = = = = = 54 = = = + = = = ( ) = − = ≈ = ≈ = 55 vi) Solving Equations with Exponents or Logarithms = = ≈ + − − = ( +− ) = = + − = + = = ± = ≠ − + + = + = = + + − = ∴ 56 = = − = − = − = − = ≈ − = − − = = = − − − = − − = − + = = = − ∴ 57 vii) Exponential Growth and Decay = ÷ ≈ ∴ 58 = + ≈ ∴ = ≈ ∴ For Practice Problems on this section, please visit: www.trentu.ca/academicskills/online_math.php 59 4. Functions and Transformations i) Definition of a Function 60 ii) Function Notation = = − − − − − = − = − − − ≠ ≠ ∴ For Practice Problems on this section, please visit: www.trentu.ca/academicskills/online_math.php 61 5. Inequalities i) Inequalities Definition ≥ ≤ ii) Linear Inequalities ≤ 62 − ∞ ii) Polynomial Inequalities (Quadratic, Cubic and Quartic Inequalities) − > − − > ∴ − ∞ − ∞ − ∞ − ∞ − ≤ + + − − ≤ − − − ∞ − [ ] [ ] ∞ − ∴ − − − 63 ⇒ ⇒ ⇒ −−∞∞ ∞∞ ∞ −−∞ − − ∴ ∴ ∴ ∞∞ ∞ − −− <<< −−−=== ⇒ ⇒ ⇒ − ∞ ((−−∞ (∞ )) ) ((∞(∞))∞ ) −−−<<< − ∞ ∴ ∴ ∴ ≠≠ ≠ ((−−∞ (∞ )) ) −− − +++>>> (( ( )()( )( )) ) −−− −−− >>> ==±±=± ==±±=± ⇒ ⇒⇒ ((−−∞(∞−−∞−)−) ) ((∞(∞))∞) ∴ ∴ ∴ ((−−∞(∞−−∞−)−)(()−− (−))(()∞(∞))∞ ) 64 Polynomial Inequalities (With Non Real Roots) ( )( ) − + + ≤ ( ) + + ≤ − ⇒ = ± ( − ∞ −] [ − ] [ ∞ ) 4 -4 ∴ For Practice Problems on this section, please visit: www.trentu.ca/academicskills/online_math.php 65 6. Trigonometric Functions i) Definitions ° ° ° −° ° + = = 66 = = = = = = = = = = = = = ° ° ° ° 67 ii) CAST Rule ° ° ° − ° ° + = + − = 68 − − ∴ + = + = ± ∴ = ± 69 iii) Radian Measure θ = θ ° π π ° π ∴ ° = π ° π = ∴ ° = π ° θ = ° ° ( π ) π ( ) − ° π − π π π −π π ( π ) = ° ( ( π ) = = ° π −π 70 π ) = −° π iv) Finding Exact Values Of Trig Ratios Special Angles And The Unit Circle π (°) π (°) π ° = = 71 θ = π ° ( ) = = × = π π π π ∴ π = − 72 π 3 π 4 π 6 π = − π − π = π/2 (0,1) ∴ π = π (−1,0)0 (1,0) ∴ π = π π − π = − 2π π π π π = π π = − −π = π = π = − π = − π π − π = π π 73 v) Using The Calculator To Find The Ratios − ° ° ° − 2π − π π π π π π = π π π 74 π π − − − − π = π π π π π − − − ° − () = ° − ° ° ≈ ° ° − − − = ° ° ° 2π − − ≈ π − 1.27 π + 1.27 π ≈ π. 75 vi) Trigonometric Equations ≤ ≤π 2π − π π π = π π 2π − − = − − = ∴ = − π π π π + − = ∴ − − = − = π ≈ ππ π π ≈ π π 2π ∴ π π 76 vii) Trigonometric Identities θ θ θ θ θ θ + θ = θ = − θ θ = − θ θ θ θ = θ θ = θ θ = θ θ = θ θ = θ θ θ 77 = ÷ ∴ = ÷ ∴ − ∴ 78 − = − + ( + ) − ( − ) ( − )( + ) ÷ − ∴ π = − = − = ∴ ∴ − − π π 79 viii) Graphs of y = sin x and y = cos x π π π π π π π π π For Practice Problems on this section, please visit: www.trentu.ca/academicskills/online_math.php 80 81 ACADEMIC SKILLS CENTRE (ASC) The Academic Skills Centre supports and empowers undergraduate and graduate students by providing flexible instruction in the skills necessary for them to succeed at university: the ability to think critically, communicate their ideas effectively, and take responsibility for their own learning. Phone (705) 748-1720 Fax (705) 748-1830 e-mail: [email protected] www.trentu.ca/academicskills
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