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Special Triangles: 45 o -45 o -90 o 1.1

Example:. 45°. 45°. 7. x. x. 45°. 45°. x. x. 7. Special Triangles: 45 o -45 o -90 o 1.1. 30°. 30°. 2 x. 2 0. 60°. 60°. x. 10. Example:. Special Triangles: 30 o -60 o -90 o 1.1. SOH – CAH – TOA. Example:. 13. 5. θ. 12.

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Special Triangles: 45 o -45 o -90 o 1.1

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  1. Example: 45° 45° 7 x x 45° 45° x x 7 Special Triangles: 45o-45o-90o 1.1

  2. 30° 30° 2x 20 60° 60° x 10 Example: Special Triangles: 30o-60o-90o 1.1

  3. SOH –CAH–TOA Example: 13 5 θ 12 Trigonometric Ratios 1.2

  4. SOH –CAH–TOA When finding angle measures use the inverse of the trigonometric functions: Example: 35 27 θ Inverse Trigonometric Ratios 1.3

  5. Examples: 1). sin 40o = cos 50o 2). cos 17o = sin 73o 3). tan 22o = 4). If sin A = 0.3, then cos B = 0.3 5). If tan B = ½, then tan A = 2 sin A = cosB cos A = sin B tan A = Trig Ratios of Complementary Angles 1.4

  6. 5 3 4 Examples: sin C = sin D = cos T = cos G = tan C = tan D = ΔDOG ~ ΔCAT sin D = sin C cos D = cos C tan D = tan C Trig Ratios of Similar Triangles 1.4

  7. Adjacent Angles: two angles in the same plane that have a common vertex, a common side, but no common interior points. Linear Pair: a pair of adjacent angles whose noncommon sides are opposite rays. Vertical Angles: two nonadjacent angles formed by two intersecting lines. Vertical angles are congruent. Special Angle Relationships Unit 2 1 and  2 are adjacent angles. 2 1 1 2  1 and  2 form a linear pair. m1 + m 2 = 180 1 2  3 and  4 are vertical angles. m  3 = m  4 3 4

  8. If two parallel lines (a and b) are cut by a transversal (c), then Corresponding Angles are : 1 3; 2  4; 5  7; 6   8 Alternate Interior Angles are : 2   7 ; 3   6 Alternate Exterior Angles are : 1  8 ; 5  4 Consecutive Interior Angles are supplementary: m3 + m2 = 180 m6 + m7 = 180 c 4 3 8 2 7 1 6 5 b a Parallel Lines

  9. Conditional statement: P Q Converse: Q P Inverse: ~ P ~Q Contrapositive: ~ Q ~P P is the hypothesis Q is the conclusion Example: If it is raining, then it is cloudy. Converse: If it is cloudy, then it is raining. Inverse: If it is not raining, then it is not cloudy. Contrapositive: If it is not cloudy, then it is not raining. Conditional Statements

  10. Law of Detachment: If P  Q is true and P is true, then Q is true. Ex: If Jim is a Texan, then he is an American. Jim is a Texan. Conclusion: Jim is an American. Law of Transitivity (syllogism): If P  Q and Q  T are true, then P  T is true. Ex: If you go to the store, then you will go to the post office. If you go to the post office, then you will buy stamps. Conclusion: If you go to the store, then you will buy stamps. Law of Contrapositive: If P  Q is true and Q is false then P is false. Ex: If two angles are complementary, then the sum of their measures equal 90o. The sum of the measures of two angles is not equal to 90o. Conclusion: The two angles are not complementary. Laws of Reasoning

  11. Circle: set of all points in a plane equidistant from a fixed point called the center. secant diameter radius chord tangent Circle

  12. Theorem 1: In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at the point of tangency. Theorem 2 : Tangent segments from a common external point are congruent. R 32 Q 3x + 5 S 50 r C r 70 32 = 3x + 5 27 = 3x 9 = x B r2 + 702 = (r + 50)2 r2 + 4900 = r2 + 100r + 2500 2400 = 100r 24 = r Properties of Tangents

  13. (1). The measure of a central angle is equal to the measure of its intercepted arc. A If G is the center of the circle and mAGB = 100o, then mAB = 100o. G B (2). The measure of an inscribed angle is one half the measure of its intercepted arc. T If R is a point on the circle and mTRS = 60o, then mTS = 120o. R S Central Angles & Inscribed Angles

  14. (1). An angle inscribed in a semicircle is a right angle. C If BC is a diameter of the circle then mCAB = 90o. A B (2). A quadrilateral can be inscribed in a circle, if and only if opposite angles are supplementary. xo + 88o = 180o and yo + 100o = 180o x = 92o y = 80o 88° 100° y° x° Inscribed Angles

  15. (2). The measure of an angle formed by 2 secants, 2 tangents, or a secant and a tangent is half the difference of the measures of the intercepted arcs. (1). The measure of the angle formed by two chords that intersect inside a circle is equal to half the sum of the measures of the intercepted arcs. Y X 80o 1 60o Z W D E 100o 40o A 1 B C Angles of a Circle

  16. (1). In the same circle, or congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. C B AB  CD if and only if D T S A (2). If one chord is a perpendicular bisector of another chord, then the first chord is a diameter. Q Since SQ  TR and SQ bisects TR, SQ is a diameter of the circle. R Properties of Chords

  17. (3). If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. If EG is a diameter and TR  DF, then HD  HF and GD  GF. F E H G (4). In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center. D C G if and only if FE  GE D E A F B Properties of Chords

  18. (2). The rule for finding segment lengths formed by two secants or a secant and a tangent is(outside)(whole) = (outside)(whole). 3 6 (1). The rule for finding segment lengths formed by two chords is (part)(whole) = (part)(whole). x 10 7 5 x 10 Circles and Segments

  19. (2). Arc Length: In a circle, the ratio of the length of a given arc to the circumference is equal to the ratio of the measure of the arc to 360o. (1). Circumference: C = 2 r or C =  d A Example: 100o 8 B Circumference and Arc Length

  20. (2). The formula for the Area of a Sectoris given by: (1). Area of a Circle: A = r2 A B Example: Let x represent the are of sector AB. 40o 8 C Area of a Circle, Area of Sector

  21. (2). Volume: (1). Surface Area: A = 4r2 Example: Find the surface area and volume of a sphere whose diameter measures 14 cm. Surface Area and Volume of Sphere

  22. vertex Lateral side altitude Base Slant height Volume: Cylinder V = πr2•h Volume: Cone l h 3 cm Volume (V) = 4 cm r Volume: Pyramid Pyramid(B = base area) The volume of a pyramid (V)= ⅓Bh Volume

  23. hB hP Volume of a Right Prism(V )= Bh (h = height of prism, B = base area) Triangular Right Prism V= Bh Volume

  24. Complex Number: Any number of the form a + bi where a and b are real numbers and i = . Since i = , then i2 = -1. Powers of i: i1 = i R = 1 i2 = -1 R = 2 i3 = - i R = 3 i4 = 1 R = 0 Example: i453 = isince the remainder is 1 when 453 is divided by 4. Complex Numbers

  25. Addition: (2 + 5i) + (3 – 4i) = (2 + 3) + (5i – 4i) = 5 +i Subtraction: (2 + 5i) – (3 – 4i) = (2 + 5i) + (-3 + 4i) = (2 + -3) + (5i + 4i) = -1 + 9i Multiplication: (2 + 5i) (3 – 4i) = 6 – 8i +15i – 20i2 = 6 + 7i – 20(-1) = 26 + 7i Operations on Complex Numbers

  26. To divide complex numbers, multiply numerator and denominator by the conjugateof the denominator. The conjugate of a + bi is a – bi. Dividing Complex Numbers

  27. Complex Number: Any number of the form a + bi where a and b are real numbers and i = . Since i = , then i2 = -1. Powers of i: i1 = i R = 1 i2 = -1 R = 2 i3 = - i R = 3 i4 = 1 R = 0 Example: i453 = isince the remainder is 1 when 453 is divided by 4. Complex Numbers 2.1A

  28. Addition: (2 + 5i) + (3 – 4i) = (2 + 3) + (5i – 4i) = 5 +i Subtraction: (2 + 5i) – (3 – 4i) = (2 + 5i) + (-3 + 4i) = (2 + -3) + (5i + 4i) = -1 + 9i Multiplication: (2 + 5i) (3 – 4i) = 6 – 8i +15i – 20i2 = 6 + 7i – 20(-1) = 26 + 7i Operations on Complex Numbers 2.2A

  29. To divide complex numbers, multiply numerator and denominator by the conjugateof the denominator. The conjugate of a + bi is a – bi. Dividing Complex Numbers 2.3A

  30. Complex Plane Imaginary axis A: 3 + 2i B: 4i C: -2 – i Real axis Absolute value of a + bi : Complex Plane/Absolute Value of Complex Number 2.4A

  31. Factoring Heirarchy: • GCF • Ex: 6x + 12y = 6(x + 2y) • Difference of Two Squares • Ex: x2 – 4 = (x + 2)(x – 2) • Factor by Grouping • Ex: 2x3 – x2 + 6x – 3 • = x2(2x – 1) + 3(2x – 1) • = (2x – 1)(x2 + 3) Factoring 2.1B

  32. Factoring Heirarchy: Trinomials with leading coefficient 1 (Use product sum method) Ex: x2 + 7x + 12 = (x + 4)(x + 3) Ex: x2 – 3x – 10 = (x – 5)(x + 2) 12 6 2 4 3 -10 -10 1 -5 2 Factoring Trinomials 2.1B

  33. Factoring Heirarchy: Trinomials with leading coefficient NOT 1 (Use adjusted product sum method/grouping) Ex: 6x2 + 7x – 3 = 6x2 + 9x – 2x – 3 = 3x(2x + 3) – 1(2x + 3) = (2x + 3)(3x – 1) 6(-3) = -18 -6 3 9 -2 Rewrite 7x as 9x– 2x Factoring Trinomials 2.2B

  34. Goal: Isolate square term on one side of the equation then take the square root of both sides. Example: 2(x + 3)2 – 5 = 11 2(x + 3)2 = 16 (x + 3)2 = 8 Add 5 to both sides Divide both sides by 2 Take square root both sides Simplify radical expression Subtract 3 from both sides Simplify radical Solving by Taking Square Roots 2.3B

  35. Goal: Set equation to zero, identify values of a, b, and c. Then, substitute values into the formula and simplify. where ax2 + bx + c = 0 Solving by Quadratic Formula 2.4B

  36. Vertex form: y = a(x – h)2 + k • Vertex: (h, k) • Axis of symmetry: x = h • If a is positive, graphs opens up, vertex is min. • If a is negative, graph opens down, vertex is max. • Extreme value: y = k • Increasing/Decreasing: Look right and left of vertex • x-Intercept: Let y = 0 and solve • y-Intercept: Let x = 0 and solve • Rate of change: Quadratic Function – Vertex Form 2.5B

  37. Standard Form: y = ax2 + bx + c • Vertex: ; then substitute to find y. • Follow remaining rules for Vertex Form 2.5B Vertex = (1, -4) Quadratic Function – Standard Form 2.6B

  38. Vertex form: y = a(x – h)2 + k Vertex = (h, k) Standard Form: y = ax2 + bx + c Vertex: Factored Form: y = a(x – r1)(x – r2) x-intercepts: r1, r2 Quadratic Functions all Forms 2.8B

  39. Arithmetic Sequence a1 is first term, d is common difference Explicit Formula: Recursive Formula: an = a1 + (n – 1)d Example: -3, 1, 5, 9, 13, . . . Explicit formula: Recursive formula: an= -3 + (n – 1)(4) Arithmetic Sequence 2.10B

  40. Arithmetic Series: The finite sum of an arithmetic sequence. Formula: Example: n = 12 a1 = 2(1) – 5 = -3 an = a12 = 2(12) – 5 = 19 Arithmetic Series 2.11B

  41. Function: x-value can NOT repeat Vertical Line Test One-to-One: x- and y-value can NOT repeat Vertical and Horizontal Line Test (2, 2), (1, -3), (0, -2) Function: Yes One-to-One: No -3 5 12 4 9 -1 0 Function: Yes One-to-One: Yes Function: No One-to-One: No Function: Yes One-to-One: No Function and One-to-One 3.2A

  42. Properties of Exponents 3.1B

  43. To solve an exponential equation, make the bases the same, then set the exponents equal to each other and solve. Example: Exponential Equations 3.4B

  44. For an exponential function of the form y = ax, If a > 1, the function is increasing throughout its domain. If 0 < a < 1, the function is decreasing throughout its domain. The asymptote has equation y = 0. There is no x-intercept. The y-intercept is (0, 1). f. End behavior: determine what happens to y as x approaches positive and negative infinity g. The rate of change for the graph is NOT constant. Exponential Function

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