2.0 CET Geometry, Trigonometry, and Probability, Lecturer's Presentation

Angles

  • Acute Angle: An angle measuring less than 90 degrees.

  • Right Angle: An angle measuring exactly 90 degrees.

  • Obtuse Angle: An angle measuring between 90 and 180 degrees.

  • Complementary Angles: Two angles whose sum is 90 degrees.

    • Example: If angles CAB and DAC are complementary, their measures add up to 90 degrees.

  • Supplementary Angles: Two angles whose sum is 180 degrees.

    • Example: If angles GEF and GEH are supplementary, their measures add up to 180 degrees.

  • Vertical Angles: Non-adjacent angles formed by intersecting lines; vertical angles are congruent (equal).

    • Example: Angles JIK and LIM are vertical angles, as are angles JIL and KIM.

  • Linear Pair of Angles: Adjacent angles whose non-common sides form a straight line. They are supplementary.

    • Example: Angles JIL and JIK, JIK and KIM, KIM and LIM, and JIL and LIM are linear pairs.

Polygons

  • Convex Polygon: A polygon in which all interior angles measure less than 180 degrees.

    • Examples: Squares and equilateral triangles.

  • Regular Polygon: A polygon that is convex, has all sides congruent (equal in length), and all angles congruent (equal in measure).

  • Number of Diagonals in a Polygon: Given by the formula:
    d = \frac{n(n-3)}{2}, where n is the number of sides.

    • Example: For a hexagon (n=6), the number of diagonals is d = \frac{6(6-3)}{2} = \frac{6(3)}{2} = 9.

  • Sum of Interior Angles of a Polygon: Given by the formula:
    Sum = (n – 2) * 180, where n is the number of sides.

Triangle

  • Altitude: A perpendicular segment from a vertex to the opposite side of the triangle.
  • Median: A segment from a vertex to the midpoint of the opposite side of the triangle, bisecting that side.
  • Angle Bisector: A segment that bisects an angle and ends on the opposite side of the triangle.
  • Acute Triangle: All angles are acute (less than 90 degrees).
  • Right Triangle: Has exactly one right angle (90 degrees).
  • Obtuse Triangle: Has exactly one obtuse angle (between 90 and 180 degrees).
  • Scalene Triangle: No two sides are congruent.
  • Isosceles Triangle: At least two sides are congruent.
  • Equilateral Triangle: All three sides are congruent.
  • Sum of Angles: The sum of the measures of the three angles of any triangle is 180 degrees.
  • Exterior Angle Theorem: An exterior angle of a triangle is greater than each of its remote interior angles. The measure of an exterior angle is equal to the sum of the measures of its remote interior angles.
  • Triangle Inequality Theorem: The sum of any two sides of a triangle is greater than its third side.
  • Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite them are also congruent.
  • Unequal Sides and Angles: The angles opposite two non-congruent sides of a triangle are not equal. The angle opposite the longer side is larger.
  • Equilateral Triangle Properties: For an equilateral triangle, the altitude, median, and angle bisector from any angle are the same segment.

Right Triangle

  • Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs).
  • Pythagorean Triples: Sets of three positive integers a, b, and c, such that a^2 + b^2 = c^2.
    • Examples: (3, 4, 5), (5, 12, 13), (7, 24, 25), (8, 15, 17), (9, 40, 41), (20, 21, 29).
    • Note: Multiples of Pythagorean triples are also Pythagorean triples (e.g., (6, 8, 10) is a multiple of (3, 4, 5)).
  • 30-60-90 Triangle: A special right triangle with angles measuring 30, 60, and 90 degrees. The sides are in the ratio 1:\sqrt{3}:2.
  • 45-45-90 Triangle: A special right triangle with angles measuring 45, 45, and 90 degrees. The sides are in the ratio 1:1:\sqrt{2}.
  • Equilateral Triangle Division: An equilateral triangle can be divided into two congruent 30-60-90 triangles by its altitude.
  • ** Triangle Similarity and Congruence**:
    • Two triangles are similar if their corresponding angles are congruent and their corresponding sides are proportional. (AA, AAA, SSS, SAS Similarity Theorems).
    • If a line parallel to one side of a triangle intersects the other two sides at distinct points, then it divides those sides proportionally.
    • The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as the third side (Midline Theorem).
    • The altitude to the hypotenuse of a right triangle divides the triangle into two smaller right triangles that are similar to each other and to the larger right triangle.
    • Two triangles are congruent if their corresponding sides and corresponding angles are congruent. (SAS, ASA, SSS, SAA Postulates).

Quadrilaterals

  • Kites: Quadrilaterals that are not parallelograms or trapezoids and have perpendicular diagonals.
  • Trapezoids: Quadrilaterals with only one pair of parallel sides.
  • Parallelograms: Quadrilaterals with two pairs of parallel sides.
  • Rhombi (Rhombuses): Parallelograms with four congruent sides.
  • Rectangles: Parallelograms with four right interior angles.
  • Squares: Rectangles with four congruent sides. They are also rhombi with four right interior angles.

Circles

  • Arc: A part of the circumference of a circle.

  • Secant: A line segment that intersects a circle in two points.

  • Tangent: A line segment that intersects a circle at exactly one point.

  • Chord: A line segment with endpoints on the same circle.

  • Diameter: A chord that passes through the center of a circle.

  • Radius: A line segment with one endpoint on the circle and one endpoint at the center; its length is half the length of the diameter.

  • Sector: A part of a circle bounded by two radii.

  • Area of a Circle: Area = \pi r^2, where r is the radius.

  • Circumference of a Circle: Circumference = 2\pi r, where r is the radius; also equals \pi d, where d is the diameter.

  • Arcs and Sectors:

    • Arc Length: The length of arc AB is given by mArcAB = 2\pi (OB) * \frac{m\angle AOB}{360°}.
    • Area of a Sector: The area of sector AOB is given by AreaSectorAOB = \pi (OB)^2 * \frac{m\angle AOB}{360°}.

  • Central Angle Theorem: The measure of a central angle is twice the measure of an inscribed angle that subtends the same arc. If O is the center of the circle then \angle BOC = 2(\angle BAC)

  • Tangent Angle Theorem: The measure of an angle formed by a tangent and a chord is one-half the measure of the intercepted arc: m\angle ABC = \frac{1}{2} (mArcADB).

  • Intersecting Secants Theorem: If two Secants intersect outside the circle then \frac{AB}{AE} = \frac{AD}{AC}

    • The measure of angle EAC is one half the difference between the measures of arcs CFE and BGD: m\angle EAC = \frac{1}{2} (mArcCFE – mArcBGD).

  • Secant Tangent Theorem: If a secant and a tangent intersect outside the circle then AB \cdot AC = AD^2

    • m\angle DAC = \frac{1}{2}(mArcCED - mArcBFD)

  • Double Tangent Theorem: If a tangents drawn outside the circle intersect at point A, then AB = AC

    • m\angle CAB = \frac{1}{2} (mArcBDC-mArcBEC)

  • Intersecting Chord Theorem: If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.

    • AE \cdot EC = BE \cdot ED
    • m\angle AEB = m\angle CED = \frac{(mArcAFB + mArcCGD)}{2}
    • m\angle BEC = m\angle AED = \frac{(mArcAID + mArcBHC)}{2}

  • Equation of a Circle:

    • Standard Form: (x – h)^2 + (y – k)^2 = r^2, where (h, k) is the center and r is the radius.
    • General Form: x^2 + y^2 + cx + dy + e = 0.

  • Transforming General Form to Standard Form: Complete the square.

    • Example: Given x^2 + y^2 – 4x + 8y + 16 = 0, complete the square to get (x – 2)^2 + (y + 4)^2 = 2^2, so the center is (2, -4) and the radius is 2.

  • Shortcut for Center: For a circle with equation x^2 + y^2 + cx + dy + e = 0, the center is \left(-\frac{c}{2}, -\frac{d}{2}\right).

Lines

  • Equation of a Line:

    • Standard Form: Ax + By + C = 0.
    • Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
    • Two-Point Form: y – y1 = \frac{y2 – y1}{x2 – x1} (x – x1).
    • Point-Slope Form: y – y1 = m(x – x1).
    • Intercept Form: \frac{x}{a} + \frac{y}{b} = 1, where a is the x-intercept and b is the y-intercept.

  • Slope of a Line:

    • If m > 0, the line leans to the right.
    • If m < 0, the line leans to the left.
    • If m = 0, the line is horizontal.
    • If m is undefined, the line is vertical.

  • Midpoint of a Line Segment: The midpoint between points (x1, y1) and (x2, y2) is \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right).

  • Finding Intersection of Lines:

    • Cancellation Method: Manipulate equations to eliminate one variable.
      • Example: Solving x + 2y = 4 and 2x + y = 8:
        -2(x + 2y) = -2(4) \Rightarrow -2x - 4y = -8
        Add this to 2x + y = 8:
        -3y = 0 \Rightarrow y = 0
        Substitute back: 2x + 0 = 8 \Rightarrow x = 4
    • Substitution Method: Solve one equation for one variable and substitute into the other.
      • Example: From x + 2y = 4, get x = 4 – 2y. Substitute into 2x + y = 8:
        2(4 – 2y) + y = 8 \Rightarrow 8 – 4y + y = 8 \Rightarrow -3y = 0 \Rightarrow y = 0
        Substitute back: x = 4 – 2(0) = 4.

  • Parallel Lines: Have equal slopes: m1 = m2.

  • Perpendicular Lines: Have slopes that are negative reciprocals of each other: m1 = -\frac{1}{m2}.

Coordinate Plane

  • Distance Formula: The distance between points (x1, y1) and (x2, y2) is given by Distance = \sqrt{(x2 – x1)^2 + (y2 – y1)^2}.

Areas and Volumes

  • Triangles:

    • Area: A = \frac{1}{2} rh, where r is the base and h is the height.

  • Equilateral Triangle:

    • Area: A = \frac{s^2\sqrt{3}}{4}, where s is the side length.

  • Rhombus:

    • Area: A = \frac{1}{2} d1 d2, where d1 and d2 are the lengths of the diagonals; also A = bh, where b is the base and h is the height.

  • Trapezoid:

    • Area: A = \frac{1}{2} (b1 + b2)h, where b1 and b2 are the lengths of the parallel sides and h is the height.

  • Sphere:

    • Volume: V = \frac{4}{3} \pi r^3, where r is the radius.
    • Surface Area: S = 4\pi r^2, where r is the radius.

  • Cylinder:

    • Volume: V = \pi r^2 h, where r is the radius and h is the height.
    • Surface Area: S = 2\pi r^2 + 2\pi rh, where r is the radius and h is the height.

  • Rectangular Box:

    • Volume: V = lwh, where l is the length, w is the width, and h is the height.
    • Diagonal: d = \sqrt{l^2 + w^2 + h^2}.
    • Surface Area: S = 2lw + 2lh + 2wh.

  • Cube:

    • Volume: V = s^3, where s is the side length.
    • Surface Area: S = 6s^2, where s is the side length.
    • Diagonal: d = s\sqrt{3}.

  • Cone:

    • Volume: V = \frac{1}{3} \pi r^2 h, where r is the radius and h is the height.

Trigonometry

Unit Circle

  • Conversion between Radians and Degrees:

    • Degrees to Radians: Angle in radians = Angle in degrees × \frac{\pi}{180°}
    • Radians to Degrees: Angle in degrees = Angle in radians × \frac{180°}{\pi}

Functions

  • Circular Functions:

    • \sin \theta = \frac{opposite}{hypotenuse}
    • \cos \theta = \frac{adjacent}{hypotenuse}
    • \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{opposite}{adjacent}
    • \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} = \frac{adjacent}{opposite}
    • \csc \theta = \frac{1}{\sin \theta} = \frac{hypotenuse}{opposite}
    • \sec \theta = \frac{1}{\cos \theta} = \frac{hypotenuse}{adjacent}
    Anglesincostan
    0°/0010
    30°/ \frac{\pi}{6}\frac{1}{2}\frac{\sqrt{3}}{2}\frac{\sqrt{3}}{3}
    45°/ \frac{\pi}{4}\frac{\sqrt{2}}{2}\frac{\sqrt{2}}{2}1
    60°/ \frac{\pi}{3}\frac{\sqrt{3}}{2}\frac{1}{2}\sqrt{3}
    90°/ \frac{\pi}{2}10undefined
    120°/ \frac{2\pi}{3}\frac{\sqrt{3}}{2}-\frac{1}{2}-\sqrt{3}
    135°/ \frac{3\pi}{4}\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}-1
    150°/ \frac{5\pi}{6}\frac{1}{2}-\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{3}
    180°/ \pi0-10
    210°/ \frac{7\pi}{6}-\frac{1}{2}-\frac{\sqrt{3}}{2}\frac{\sqrt{3}}{3}
    225°/ \frac{5\pi}{4}-\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2}1
    240°/ \frac{4\pi}{3}-\frac{\sqrt{3}}{2}-\frac{1}{2}\sqrt{3}
    270°/ \frac{3\pi}{2}-10undefined
    300°/ \frac{5\pi}{3}-\frac{\sqrt{3}}{2}\\frac{1}{2}-\sqrt{3}
    315°/ \frac{7\pi}{4}-\frac{\sqrt{2}}{2}\frac{\sqrt{2}}{2}-1
    330°/ \frac{11\pi}{6}-\frac{1}{2}\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{3}
    360°/ 2\pi010

Identities

  • Pythagorean Identities:

    • \cos^2 \theta + \sin^2 \theta = 1
    • 1 + \tan^2 \theta = \sec^2 \theta
    • \cot^2 \theta + 1 = \csc^2 \theta

  • Law of Sines:

    • \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

  • Law of Cosines:

    • a^2 = b^2 + c^2 – 2bc \cos A
    • b^2 = a^2 + c^2 – 2ac \cos B
    • c^2 = a^2 + b^2 – 2ab \cos C

  • Cofunction Identities:

    • \sin(\frac{\pi}{2} – x) = \cos x
    • \cos(\frac{\pi}{2} – x) = \sin x
    • \tan(\frac{\pi}{2} – x) = \cot x
    • \cot(\frac{\pi}{2} – x) = \tan x
    • \sec(\frac{\pi}{2} – x) = \csc x
    • \csc(\frac{\pi}{2} – x) = \sec x

  • Even-Odd Identities:

    • \sin(-x) = -\sin x
    • \cos(-x) = \cos x
    • \tan(-x) = -\tan x
    • \cot(-x) = -\cot x
    • \sec(-x) = \sec x
    • \csc(-x) = -\csc x

  • Period Identities:

    • \sin(x + 2k\pi) = \sin x
    • \cos(x + 2k\pi) = \cos x
    • \tan(x + k\pi) = \tan x
    • \cot(x + k\pi) = \cot x
    • \sec(x + 2k\pi) = \sec x
    • \csc(x + 2k\pi) = \csc x

  • Angle Addition/Subtraction Identities:

    • \cos(\alpha + \beta) = \cos \alpha \cos \beta – \sin \alpha \sin \beta
    • \cos(\alpha – \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta
    • \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
    • \sin(\alpha – \beta) = \sin \alpha \cos \beta – \cos \alpha \sin \beta
    • \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 – \tan \alpha \tan \beta}
    • \tan(\alpha – \beta) = \frac{\tan \alpha – \tan \beta}{1 + \tan \alpha \tan \beta}

  • Double Angle Identities:

    • \sin 2\alpha = 2 \sin \alpha \cos \alpha
    • \cos 2\alpha = 1 – 2 \sin^2 \alpha = 2 \cos^2 \alpha – 1
    • \tan 2\alpha = \frac{2 \tan \alpha}{1 – \tan^2 \alpha}
    • \cot 2\alpha = \frac{\cot^2 \alpha – 1}{2 \cot \alpha}

  • Half Angle Identities:

    • \sin(\frac{\alpha}{2}) = \pm \sqrt{\frac{1 – \cos \alpha}{2}}
    • \cos(\frac{\alpha}{2}) = \pm \sqrt{\frac{1 + \cos \alpha}{2}}
    • \tan(\frac{\alpha}{2}) = \frac{1 – \cos \alpha}{\sin \alpha} = \frac{\sin \alpha}{1 + \cos \alpha}
    • \cot(\frac{\alpha}{2}) = \frac{1 + \cos \alpha}{1 – \cos \alpha}

  • Product to Sum Identities:

    • \sin \alpha \sin \beta = \frac{1}{2} [\cos(\alpha – \beta) – \cos(\alpha + \beta)]
    • \cos \alpha \cos \beta = \frac{1}{2} [\cos(\alpha – \beta) + \cos(\alpha + \beta)]
    • \sin \alpha \cos \beta = \frac{1}{2} [\sin(\alpha + \beta) + \sin(\alpha – \beta)]
    • \cos \alpha \sin \beta = \frac{1}{2} [\sin(\alpha + \beta) – \sin(\alpha – \beta)]

  • Sum to Product Identities:

    • \sin \alpha + \sin \beta = 2 \sin(\frac{\alpha + \beta}{2}) \cos(\frac{\alpha – \beta}{2})
    • \sin \alpha – \sin \beta = 2 \cos(\frac{\alpha + \beta}{2}) \sin(\frac{\alpha – \beta}{2})
    • \cos \alpha + \cos \beta = 2 \cos(\frac{\alpha + \beta}{2}) \cos(\frac{\alpha – \beta}{2})
    • \cos \alpha – \cos \beta = -2 \sin(\frac{\alpha + \beta}{2}) \sin(\frac{\alpha – \beta}{2})

Inverse Trigonometric Functions

  • Inverse Trigonometric Functions:

    • \arcsin \theta = \sin^{-1} \theta = inverse of sine. Ex. \arcsin 0 = 0 since \sin 0 = 0
    • \arccos \theta = \cos^{-1} \theta = inverse of cosine. Ex. \arccos \frac{\sqrt{2}}{2}=0 since \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}
    • \arctan \theta = \tan^{-1} \theta = inverse of tangent. Ex. \arctan 1=0 since \tan \frac{\pi}{4} = 1
    • \arccot \theta = \cot^{-1} \theta = inverse of cotangent. Ex. \arccot \sqrt{3} = \frac{\pi}{6} since \tan \frac{\pi}{6} = \sqrt{3}
    • \arcsec \theta = \sec^{-1} \theta = inverse of secant. Ex. \arcsec \sqrt{2} = \frac{\pi}{4} since \sec \frac{\pi}{4} = \sqrt{2}
    • \arccsc \theta = \csc^{-1} \theta = inverse of cosecant. Ex. \arccsc 2 = \frac{\pi}{6} since \csc \frac{\pi}{6} = 2

Statistics and Probability

Sigma Notation

  • Sigma Notation:
    \sum_{x=m}^{n} F(x) = F(m) + F(m + 1) + … + F(n)

  • Properties of the Sigma Notation:

    • \sum{x=m}^{n} CF(x)=c\sum{x=m}^{n} F(x), where c is any constant
    • \sum{x=m}^{n} F(x) + G(x) = \sum{x=m}^{n} F(x) + \sum_{x=m}^{n} G(X)
    • \sum_{x=m}^{n} c = c(n-m+1)

Measures of Central Tendency

  • Mean (average): The sum of all data values divided by the population/sample size.
    \text{mean} = \frac{\sum x}{n}, where n = population/sample size.
  • Median: The positional center of an arrayed (arranged from lowest to highest) data
    \text{median} = \frac{n+1}{2}, when n is odd,
    \text{median} = \frac{x{\frac{n}{2}} + x{\frac{n}{2}+1}}{2}, when n is even
  • Mode: The data element that occurs most often. A set with more than one mode is called multimodal.

Measures of Variability

  • Range - difference between highest value and lowest value
  • Population Variance - \sigma^2 = \frac{\sum{i=1}^{N} (xi - \mu)^2}{N}
  • Sample Variance - s^2 = \frac{\sum{i=1}^{n} (xi - \bar{x})^2}{n-1}
  • Standard Deviation - square root of variance

Measures of Position

  • Percentile: A formula for the kth percentile, Pk for a data set with n elements is given by the following:

    • Pk = \begin{cases} X{\lceil{\frac{nk}{100}}\rceil} & \text{if } \frac{nk}{100} \text{ is an integer} \ \frac{X{\lfloor{\frac{nk}{100}}\rfloor} + X{\lfloor{\frac{nk}{100}}\rfloor+1}}{2} & \text{otherwise} \end{cases}

    • n = number of observations.

    • k = percentile.

    • [[x]] is the greatest integer function. This function simply gets the integer portion of a number.

      • [[1.2]] = 1
      • [[1.9]] = 1
      • [[2.1]] + 1 = 2 + 1 = 3

  • Equivalence of Percentiles (Pk), Quartiles (Qk), and Deciles (Dk):

    • Quartile
      • Q1 = P25
      • Q2 = P50
      • Q3 = P75
    • Decile
      • D1 = P10
      • D2 = P20
      • D3 = P30
      • D4 = P40
      • D5 = P50
      • D6 = P60
      • D7 = P70
      • D8 = P80
      • D9 = P90
    • Also note that Median = D5 = Q2 = P50 so in case you forget the formula for the median, you can derive it using the formula of the 50th percentile.

Combinatorics

  • Fundamental Counting Principle: If there are a possible outcomes of event A, and b possible outcomes of event B, the number of possible outcomes of event A and event B is a multiplied by b.

  • Factorial: n! = n x (n-1) x (n-2) x … x 2 x 1

  • Permutation (order matters):

    • P(n, r) = \frac{n!}{(n-r)!}

      • n = total number of elements in set
      • r = number of chosen elements

  • Circular permutation of n objects = (n – 1)!

  • Keyring permutation of n objects = \frac{(n – 1)!}{2}

  • Permutation with repeating objects:
    \frac{n!}{r1!r2!…rk!} * where n is the total number of objects, r is the number of objects taken at a time, r1, r2, r3…, r_k are the numbers of objects that are repeated

  • Combination (order does not matter):

    • C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!}

      • n = total number of elements in set
      • k = number of chosen elements

Probability

  • P(A) = (no. of possible ways A can happen) / (no. of all possible outcomes)

  • P(A) + P(A') = Probability of A happening + Probability of A not happening = 1

  • P(A or B) = P(AUB) = P(A) + P(B) – P(A∩B) where A and B are events in a sample space

  • If A & B are mutually exclusive events, then P(AUB) = P(A) + P(B)

  • Under independence of A & B, P(A∩B) = P(A)P(B)

  • Under condition of B, P(A∩B) = P(A|B)P(B)

  • Dependent events – events that affect each other’s probability

    • Example: Committing a crime affects the probability of