Kindergarten Maths 幼儿园数学入门

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Study notes mainly from Khan Academy. Not literally for Kindergarten maths only, but also including all K12 math subjects.

KHAN ACADEMY MASTERY rules collected:

• 1 skill means 1 practice subject, could be found at practice page on the subject.
• 1 skill includes 4 to 7 questions.
• 1 skill finished 75%, will promote to Practiced.
• 1 skill finished 100%, will promote to LEVEL ONE.
• 1 skill must be practiced to join the Mastery challenge.
• 1 question correct on Mastery challenge, the related skill will get 1 level up.
• 1 question incorrect on Mastery challenge, the related skill will get 1 level down.
• 1 quiz finished in course section, the correct skills will promote to LEVEL ONE.
• 1 unit test finished in course section, the correct skill will get 1 level up till MASTERY. Incorrect answer will get 1 level down.

Practice To-do List

• Algebra basics
• 7th grade
• 8th grade
• Algebra 1
• Algebra 2
• Geometry
• Trigonometry
• High school statistics
• Precalculus
  • Trigonometry
  • Conic sections
  • Vectors
  • Matrices
  • Complex numbers
  • Probability and combinatorics
  • Series

Khan Academy Mastery challenge

• Early Math (100%)
• Kindergarten (100%)
• 1st grade (100%)
• 2nd grade (100%)
• 3th grade (100%)
• 4th grade (100%)
• 5th grade (100%)
• 6th grade (100%
• 7th grade (100%)
• 8th grade (100%)
• Arithmetic (100%)
• Basic geometry (100%)
• Pre-algebra (100%)
• Algebra basics (100%)
• Algebra 1 (100%)
• Algebra 2 (100%)
• Geometry (100%)
• Trigonometry (100%)
• High school statistics (100%)
• Precalculus (100%)
• Mathematics 1 (100%)
• Mathematics 2 (100%)
• Mathematics 3 (100%)
• Linear Algebra (No mastery) (100%)
• AP calculus AB
• AP Statistics (No mastery)

Unit Tests (For reviewing forgettable High school level concepts)

Algebra (All contents):

• Linear equations, functions, & graphs (Slope, Linear equation forms)
• Sequences (arithmetic/geometric sequences)
• Functions (Domain, Range, Interval, ARC, Transformation, Inversion, Symmetry, End behavior, Periodicity)
• Quadratic equations & functions (Parabola, Quadratic forms, Transform quadratic functions)
• Polynomial expressions, equations, & functions (monomial, binomial, Zeros of polynomial func)
• Exponential & logarithmic functions (Exponent properties, Radicals, growth&decay, Properties of logarithms)
• Radical equations & functions (Extraneous solutions, Domain of radical functions)
• Rational expressions, equations, & functions (rational expressions, End behavior, Discontinuities)
• Trig functions (Radians, Graphs, Sinusoidal functions, Inverse, Trig identities)
• Complex numbers (Absolute value & angle, Distance & midpoint, Forms of complex numbers)
• Conic sections (Functions of conic shapes, Foci, Hyperbola, Ellipse)

Geometry:

Trigonometry:

Statistics:

Review hardcore quiz

• Trigonometry
  • Sinusoidal functions
  • Graph sinusoidal functions
  • Inverse trig functions
• Probability
  • Probability basics
  • Permutations and combinations
• Algebra 2
  • Functions
  • Quadratics & Parabola
  • Polynomials
  • Graphs of rational functions
  • Exponential and Logarithmic functions
  • Series
• Precalculus
  • Conversion of complex numbers

Table of Contents

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How To Read Math Notations

Collect some common maths notions.

Refer to wiki: List of mathematical symbols

For further information, review this link, or download this pdf: Guidelines for Reading Mathematics.pdf.
Notice that: Word in PARENTHESIS doesn't have to say out.

Product

"Three times four"
"Three multiply by four".

3 × 4

"Open parenthesis x plus three close parenthesis multiply by open parenthesis ...."
"x plus three, multiply by x minus two."

(x+3)(x-2)

Quotient & Fraction

"Three fourth "
"Three over four".

3/4

"One half"
"One over two"

1/2

"Three halves"
"Three over two".

3/2

"x plus one over x minus one" .

(x+1)/(x-1)

exponent

"x squared"
"x (raised) to the second (power)"

x²

"x cubed"
"x to the third (power)"

x³

"Three to the zeroth power"

3⁰

"Nine to the a plus b (power)."

9⁽ª⁺ᵇ⁾

"Five to the three t (power)."

5³ᵗ

Factorial !

"4 factorial"
"4 shriek"
"4 bang"

4!

composite function

"f of x."

f(x)

"g of f of x."

(g◦f)(x)

"h of g of minus six".

(f◦g)(-6)

Set

"x in B", "x belongs to B", or "x is an element of B"

x ∈ B

"y not in B", or "y does not belong to B"

y ∉ B

Subset

"A is a subset of B", or "A is contained in B"

A ⊇ B

"B is a superset of A", or "B includes A", or "B contains A"

B ⊇ A

The relationship between sets established by ⊆ is called inclusion or containment.

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How to use calculator

Latex syntax

These syntax works for Chrome address bar, Mac Spotlight bar, Cymath etc.

• Exponent: for 3⁴, type 3^4, results 81.
• Root: for √9, type 9^(1/2), results 3.
• Logarithm: for log₂8, type log_2(8), results 3.
• Logarithm base-e: for ln 6, type ln 6, results 1.791....
• Logarithm base-10: for log 10, type log 10, results 1.

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Triangle types

Refer to Khan academy: Triangle types

Types by side length

• Scalene triangles with all different sides （sounds "scale-lin")
• Isosceles triangles with 2 equal sides (sounds "i-saw-sillys")
• Equilateral triangles with 3 equal sides (sounds "e-qui-lateral")

Types by angles

• Acute triangles with 3 acute angles
• Obtuse triangles with 1 obtuse angle
• Right triangles with 1 right angle.

Angle types

Types of angles: acute, right, obtuse(sounds "ob-tuse"), and straight.

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Dividing numbers strategy

Khan page.

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Polyhedra - 3D shape

Polyhedra, or Polyhedrons is plural for Polyhedron. means a 3D shape with surfaces all flat.
Examples: Cube, Triangular Prism, Pyramids, Tetrahedron

More about the list of polyhedrons: Animated Polyhedron Models

Refer to math is fun: Polyhedron

Refer to Khan academy lession.

Regular polyhedrons:

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Prisms (3D shape)

A Prism is a 3D shape, which could be stretch out from a 2D shape with sides all straight. The 2D shape is called Cross-Section.
A prism is a polyhedron, which means all faces are flat!

Refer to math is fun: Prisms - 3D shape

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Mean, median, & mode numbers (STATS)

Khan lecture.

• Mean number is just an average of all numbers listed.
• Median number is the middle positioned number in a ordered number set (means no duplicates). If there're two middles, then average them to get a median number.
• Mode number is the number shows up most times in a list.

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Dependent variables & independent variables

Refer to math is fun: dependent variable
Refer to math is fun: independent variable

Understand:

• Independent variables: Input value of a function. (usually x)
• Dependent variables: Output value of a function. (usually y)

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Box plots and Quartiles (STATS: Distribution graph)

It's also called Box and whisker plots, or Five-number summary.

Khan lecture 1.
Khan lecture 2.
Maths is for fun Wiki.

Five-number summary

Interquartile range (IQR) (STATS: Box plot)

Refer to Khan academy.

Example

Example

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Clusters, gaps, peaks & outliers (STATS: Distribution graph)

Khan lecture: Shape for distributions.
Khan lecture 2 Clusters, gaps, peaks & outliers.

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What are sine, cosine, tangent? (TRIGONOMETRY)

• Khan short video
• Maths is Fun

First to know the terms

It's only applicable to an Right triangle. A right triangle is a triangle with a 90-degree angle.
The angle we pick out, is called Θ, theta. In vector related problems, it's also called the direction angle, see this Khan practice.

The side opposite to that angle is called Opposite.
The side next to the angle is called Adjacent, "A-Jason-t".
The side is long is called Hypotenuse, "High-po-ten-news`.

How to calculate sine, cosine and tangent of an angle

REMEMBER SHORTCUT: SOH-CAH-TOA, pronounced "so-kah-tow-ah".

• Sin = Opposite / Hypotenuse
• Con = Adjacent / Hypotenuse
• Tan = Opposite / Adjacent

How to calculate the ratio of an angle?

We want to know what the ratio of an angle is, 60° or 59°?
If we know tan(x) = 123, whatever, then we can get it by x = tan_inverse(123).
tan_inverse(...) or tan⁻¹(...) are the same.

Reciprocal trig ratios: Cosecant, Secant, Cotangent

Khan notes.

• Cosecant: csc(θ) = 1 / sin(θ)
• Secant: sec(θ) = 1 / cos(θ)
• Cotangent: cot(θ) = 1 / tan(θ)

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Terms, factors, & coefficients in Algebra

Khan lecture.

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Mean absolute deviation (MAD) - STATS

The deviation is the distance from the value to the mean value.
It's used to describe how the values looks like or how they're laid on the axis, are they close to each other or far away.

The Mean absolute deviation is the absolute average of all deviations.

Khan lectures.

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Common 2D shapes

Quadrilaterals

Polygons

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Distance between points & lines

Distance of two points on x-y axis?

Refer to Khan academy lecture: pythagorean-theorem-distance

▼ Just to apply the Pythagorean Theorem:

Example

What is the distance between (-6, 8) and (-3, 9) ?
Solve:

• Easy way to do this:
• Make a triangle with two points same with the given points, and solve the length of the triangle.
  
• So to solve the example, just like this:
  

Distance between point & line

Refer to Khan academy: Distance between point & line

Slope of a perpendicular line is the Negative Inverse of the slope of the given line.

Strategy:

• Draw a perpendicular line form the point to the given line.
• Find out the linear equation for this perpendicular line by:
  • Find the slope: which is the Negative Inverse of the slope of the given line:
    
  • Find the shift: plug in the point's x & y coordinates and get the shift
• Set two lines' equations equal and get the Intersect point's coordinates.
• Calculate two points distance by Pythagorean Theorem:
  

Example

Solve:

• Find the perpendicular line from the point to the given line:
  • Slope = -1/-1 = 1
  • y=ax + b -> 2 = -3*1 +b -> b=5 -> y=x+5
• Find the intersect: -x+1 = x+5 -> x=-2 -> y=3 -> intersects at (-2, 3)
• Calculate distance between two points:
  
• The answer is √2

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Makeup price problem

A markup rate is a percentage of the wholesale price that a store adds to get a selling or retail price.

Question

Answer

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Pythagorean Theorem (TRIANGLE)

勾股定理。
Pronounced: Pe~'tha-gor-rean 'Theor-rem
In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.

Refer to Math is fun: Pythagorean theorem

Triangle inequality theorem

Refer to Math is fun: Triangle inequality theorem

In a triangle, the sides always follow these two rules:

• Any side of a triangle is always shorter than the sum of the other two sides
• The third side must be also larger than the difference between the other two sides

Example:

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Complementary, supplementary, vertical angles

Refer to Khan academy: Complementary, supplementary, vertical angles

Complementary angles are two angles with a sum of 90°. A common case is when they form a right angle.

Supplementary angles are two angles with a sum of 180°. A common case is when they lie on the same side of a straight line.

Vertical angles are angles opposite each other where two lines cross. A pair of vertical angles have the same measure of angle.

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Transformations (GEOMETRY)

Transformation means something's changing, transforming.

Refer to Khan academy: Transformations

Types of transformations:

• Translations: Slide or move the shape. In geometry, a translation moves a thing up and down or left and right.
• Rotation: Turn or rotate the shape. In geometry, rotations make things turn in a cycle around a definite center point.
• Reflection: Flip or mirror the shape. A reflection is a transformation that acts like a mirror: It swaps all pairs of points that are on exactly opposite sides of the line of reflection.
• Dilation: Expand or shrink the shape.

And,
you can group above types of transformations into two groups: Rigid transformations and Dilations

Rigid transformations and Congruent

Rigitd transformations means you play around the shape without expanding or shrinking it.
Or say, without dilation, all translation/rotation/reflection would be rigid transformation.

Congruent is the shape after you "rigid transformed`.
Or say, without dilation, after all translation/rotation/reflection, the shape is called "congruent" to the original shape.

Khan lecture.

Dilations

Dilation, just a fancy word for "resizing", or "scaling".

Notice that: Dilation DOES NOT change any angle within the shape.
Don't get confused with a horizontal stretch, which does change both sides and angles.

Similarity

When you resize, or say dilate a shape, you call the new shape similar to the original one.
If nothing changed with the shape, you call it congruent to the original one.

Scale factor

Means how much you scale the shape, like 2 times bigger, or 2/3 smaller.

Notice:

• The scale factor is of the length of the shape, NOT the area of it.

Scale factors and area

When you scale the shape, the area of the new shape is (scale facto)² times to the original one.

Khan lecture: Scale factors and area

For Shape A and scaled shape A', it leads to two practical conclusions:

• If we know the scale factor is x, then the area of A is x² times to the original one.
• If we know the ratio (area of A') ÷ (area of A) is x, then the scale factor is √x

Example

Solve:

• New area G is 1/9 of F
• So the (scale factor)² = 1/9, which results the scale factor = 1/3

Dilation Center

"Dilate the shape ABOUT a point P", means take the point as a center to dilate the shape.

How does it work? As the picture below, just simply scale the distance from each vertex(point) of the shape to the point.

How to find the dilation center?

The point P and it's image and dilation center, should be IN ONE LINE !

In the example below, you should forget about the origin but set the P point as origin and count the distance of each vertex of the triangle:

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Area, perimeter, surface, volume of shapes (Geometry)

Cheatsheets for calculating area, perimeter, surface, volume of common shapes.

Area of plane shapes

Triangle, square, rectangle, parallelogram, trapezoid, circle, ellipse, sector.

Refer to math is fun: area

Areas:

• Area of Equilateral Triangles:
  
• Isosceles Right Triangle:
  

Surface Area

• Surface of Cylinder: 2π · r · (r + h)
• Surface of Sphere: 4π · r²
• Surface of Cone: π · r( r + √(h²+r²) )

Volume of Cuboids, Rectangular Prisms and Cubes

Volume = Length · Width · Height

Volume of Cone, sphere, Cylinder

The volume of a Cylinder is: π · r² · h
The volume of a Cone is: 1/3 π · r² · h
The volume of the Sphere is:4/3 π · r³

Refer to math is fun: cone-sphere-cylinder

Volume, surface area of Pyramid

Refer to math is fun: pyramids

• The Volume of a Pyramid: 1/3 · [Base Area] · Height
• The Surface Area of a Pyramid:
  • When all side faces are the same: [Base Area] + 1/2 Perimeter · [Slant Length]
  • When side faces are different: [Base Area] + [Lateral Area]

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Rational & irrational numbers

A rational number can always be expressed as a fraction of two integers.
vice versa, a irrational number cannot be expressed as a fraction of two integers.

Refer to math is fun: Rational & irrational numbers

A square root of a non-perfect square is an irrational number, because it cannot be expressed as the fraction of two integers.

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Stem and Leaf Plots (Stats)

A Stem and Leaf Plot is a special table where each data value is split into a "stem" (the first digit or digits) and a "leaf" (usually the last digit).

Refer to khan academy: Stem and Leaf Plots
Khan lecture.

In the above plot:

• The stem column means the tens number of the whole number.
• The row column means the ones number of the whole number.
  For example, the numbers of the second row represent: 12, 13, 15.

How the Stem and Leaf Plot represent numbers, it depends on the context. Stem can represents number of tens place, or ones place; Leaf can represents ones place or even decimal places.

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Proportional relationship

Linearly, if x and y has a relationship that one is another's proportion, then they have a proportional relationship. Expression as:
y/x = k.

Or it's a version of linear equation y=mx +b, only the b=0, then y=mx.

A relationship is proportional if its graph is a straight line through the origin. Remember that the origin is the point (0,0)

At the showing image above, only first one shows a proportional relationship. The other two are not linear and going through origin.

Directly Proportional & Inversely Proportional

Refer to Maths is fun.

• Directly proportional: As one amount increases, another amount increases at the same rate.
  
• Inversely Proportional: when one value decreases at the same rate that the other increases.
  

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Three Terms ratio problem

It takes 54 minutes for 4 people to paint 6 walls.
How many minutes does it take 6 people to paint 7 walls?

• 4 p 1 wall: 54 / 6 = 9 minutes, then:
• 1 p 1 wall: 9 * 4 = 36 minutes, then:
• 1 p 7 wall: 36 * 7 = 252 minutes, then:
• 6 p 7 wall: 252 / 6 = 42 minutes.

Note that, it cannot be easily get out the result, but only to understand the context and make it out step by step.

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Scientific Notation

Scientific Notation (also called Standard Form in Britain) is a special way of writing numbers:
It makes it easy to use big and small values.

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The Slope of a line

Refer to math is fun: slope

• Rise: the vertical change is called "rise".
• Run: the horizontal change is called "run".

• Positive slope: the line is going up.
• Negative slope: the line is going down.
• Slope of zero: a horizontal line.
• Undefined slope: a vertical line.

Slope-intercept form

y = mx +b is called "slope-intercept form` of an equation.

• m is the slope of this line
• b is the y-intercept of this line.

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Polynomials

Refer to math is fun: Polynomials

Just an expression of some values, including variables, exponents, constants...
It only fits to "poly" "nomials", aka. multiple terms, as below:

Degrees of a polynomial

Means the highest exponent of a variable in the polynomial.

For example, if it's a quadratic, then it's a 2nd degree polynomial.

Monomial, Binomial, Trinomial

Multiply binomials

Refer to math is fun: special-binomial-products

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Quadratics

Refer to math is fun: Quadratics

Different forms a quadratic

There're many tricks to convert one form to another, mainly depends on what information we have and what information we'd like to get.

Common forms for quadratic as below:

• Standard form:
• Two binomials form: Could get two solutions without calculator.
• Perfect square form: Could get the positions of vertex by only eyeballing it. In another way, you could tell how much the graph was moved from the origin point.

What is the Solutions to the quadratic

The solutions to the Quadratic Equation are where it is equal to ZERO.

They are also called ROOTS, or sometimes ZEROS, or REAL ZEROS, OR REAL NUMBER ZEROS, or DISTINCT REAL NUMBER ZEROS.
Graphically, when the graph touches x-axis, the point is A SOLUTION to the quadratic equation.

There're different ways to get solutions for the quadratic:

• Factor the quadratic as a product of two binomials
• Re-group the quadratic, and then factor it
• Use Quadratic Formula.

To factor a quadratic as the product of two binomials

To find out the answer quickly and get intuition of it step by step, visit this cheat online solver: Cymath.

It's easy to expand product of two binomials to a quadratic, but tricky to factor them back.

What we should do is, in the standard quadratic form ax² + bx +c = 0, we are to:

• Find the two common factors of "c"
• Be sure the sum of two factors equals to "b"
  As the example below, factors of "c" could be 4 and 6, and the sum of them is 10. So the answer is (x+4)(x+6)
  

To factor quadratics by re-grouping (to factor more complicated quadratics)

Refer to khan academy: factoring-by-grouping

When the coefficient of the 2nd degree term is not 1, things get more complicated.

Khan notes 1.
Khan notes 2.

The way to do it is, in the standard form of Ax²+Bx+C:

• Find two numbers add up equal B
• The product of the two numbers should equal to A×C
• Regrouping Bx to two terms of x with the two number.

Use Quadratic formula to solve equation

Refer to math is fun: quadratic-equation-derivation

If we can't easily factor the quadratic, we have to use the ultimate formula:

It's an universal method for solving any quadratic equation.

Discriminant of a quadratic

Refer to khan academy: discriminant-review

Within the quadratic formula, there is a b2 − 4ac called Discriminant.

Refers to Mathwarehouse.

The solutions table as below:

The meaning of it, is to tell us about the solution of a quadratic:

• when b2 − 4ac is POSITIVE, we get TWO Real solutions. (It touches x-axis twice.)
  

• when it is ZERO we get just ONE Real solution. (Its vertex is on x-axis.)
  

• when it is NEGETIVE we get TWO Complex solutions. (It won't touch x-axis at all.)
  

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Circle

"The circle is arguably the most fundamental shape in our universe." - Sal Khan

Arc length

radius × θ = arc length

Inscribed Angles

• Inscribed angle's notation: ψ, called "si", pronounced "sai".
• Central angle's notation: θ, called "theta", pronounced "thay-ta".
• Intercepted arc's notation: 𝞪, called "alpha".

Inscribed Angle Theorem

For all inscribed angles, θ = 2ψ.

Tangent line

A line that is tangent to a circle at a particular point is perpendicular to the radius at that point.

Two segments tangent to a circle from a common endpoint are congruent, aka. having the same length.

Circumscribed angle

Angle A is circumscribed about circle O.

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parallel & perpendicular lines equations

• Two lines are parallel: slopeA = slopeB
• Two lines are perpendicular: slopeA = -1 / slopeB

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Specifying planes in three dimensions

A plane could be defined by 3 NON-COLINEAR points .

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All trig identities

Refer Khan notes.
PDF: (Trigonometry) Trig Formulas.pdf

Reciprocal and quotient identities

tan(θ) = sin(θ) / cos(θ)

csc(θ) = 1/sin(θ)
sec(θ) = 1/cos(θ)
cot(θ) = 1/tan(θ)

Pythagorean identities

sin²(θ) + cos²(θ) = 1

cos²(θ) - sin²(θ) = cos(2θ)

sec²(θ) - tan²(θ) = 1

csc²(θ) - cot²(θ) = 1

?? identities

sin(θ) = sin(π - θ)    #  if θ is POSITIVE.
sin(θ) = sin(-π - θ)   #  if θ is NEGATIVE.

cos(θ) = cos(-θ)

tan(θ) = tan(-π + θ)   #  if θ is POSITIVE.
tan(θ) = tan(π + θ)    #  if θ is NEGATIVE.

Symmetry and periodicity identities

sin(-θ) = -sin(θ)
cos(-θ) = cos(θ)
tan(-θ) = -tan(θ)

sin(θ) = sin(2π+θ)
cos(θ) = cos(2π+θ)
tan(θ) = tan(π+θ)

Angle Addition & Subtraction identities

sin(a+b) = sin(a)cos(b) + cos(a)sin(b)
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)

cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
cos(a-b) = cos(a)cos(b) + sin(a)sin(b)

tan(a+b) = ( tan(a)+tan(b) ) / ( 1-tan(a)tan(b) )
tan(a-b) = ( tan(a)-tan(b) ) / ( 1+tan(a)tan(b) )

Double angle identities

sin(2θ) = 2·sin(θ)cos(θ)

cos(2θ) = cos²(θ) - sin²(θ)
cos(2θ) = 2·cos²(θ) - 1

tan(θ) = 2·tan(θ) / ( 1-tan(θ) )

Cofunction identities

sin(θ) = cos(π/2 - θ)
cos(θ) = sin(π/2 - θ)
tan(θ) = cot(π/2 - θ)

cot(θ) = tan(π/2 - θ)
sec(θ) = csc(π/2 - θ)
csc(θ) = sec(π/2 - θ)

Half angle identities

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Inverse Trig equations

Probably it's the most tricky and difficult topic throughout all High School Math.

Prerequisites:

• Unit circle
• Reference angle & Angle in standard position
• Inverse functions
• Radian & degree conversion
• Signs of trig functions

It asks you to solve a trig function like this:

Actually it's asking you to use inverse trig function skills.

Multiple solutions for inverse trig function

The most difficult part to understand is the multiple solutions. And not all are valid.

e.g. sin⁻¹(1/2) means, we know the sine value is 1/2, and we'd like to get the arc angle's measure. But there're multiple arc angles could have the very SAME sine value:

• Counter-clockwise: 30°, 150°
• Clockwise: -330°, -210°.

But for making the function valid, we have to make it to 1-INPUT-1-OUTPUT.
So we are to filter them out and only keep one solution, by restrict the angle's measure.

The angle's measure in Sine function is restricted by DOMAIN, but in its inverse function, it became RANGE.

By filtering with the RANGE, we are to get a ONLY-ONE & MUST-ONLY-ONE answer.

Principle value & Calculator

So after filtering out all other solutions, we have only ONE solution, we call it:
The Principle Value of inverse trig function,
which shows up in any calculator when you type it.

HOW TO GET ALL POSSIBLE SOLUTIONS

• First to use a CALCULATOR to get the principle value as our base solution.
• Find the mirror solution of each trig function by Trig symmetry identities:
  • sin(θ) = sin(π - θ), if θ is POSITIVE.
  • sin(θ) = sin(-π - θ), if θ is NEGATIVE.
  • cos(θ) = cos(-θ)
  • tan(θ) = tan(-π + θ), if θ is POSITIVE.
  • tan(θ) = tan(π + θ), if θ is NEGATIVE.
• Add periodicity to the solution to represent all periodic solutions: x = θ + 2πn

Example: sin(x) = 0.65

• Use calculator to do arcsin(0.65) and get principle value: 0.71 Rad.
• Apply trig identity sin(θ) = sin(π-θ) to get the mirror solution: arcsin(0.65) = π - arcsin(0.65), which'd be: 2.43 Rad.
• With two solutions 0.71 Rad and 2.43 Rad and add periodicity, we'll get two full solutions:
  • x = 0.71 + 2πn
  • x = 2.43 + 2πn

Example: sin(x) = −0.25

• Use calculator to do arcsin(-0.25) and get principle value: -0.25 Rad.
• Apply trig identity sin(θ) = sin(-π-θ) to get the mirror solution: arcsin(-0.25) = -π - arcsin(-0.25), which'd be: -2.89 Rad.
• With two solutions -0.25 Rad and 3.39 Rad and add periodicity, we'll get two full solutions:
  • x = -0.25 + 2πn
  • x = -2.89 + 2πn

Example: cos(x) = −0.7

• Use calculator to do arccos(-0.7) and get principle value: 2.35 Rad.
• Apply trig identity cos(θ) = cos(-θ) to get the mirror solution: arccos(-0.7) = - arcsin(-0.7), which'd be: -2.35 Rad.
• With two solutions 2.35 Rad and -2.35 Rad and add periodicity, we'll get two full solutions:
  • x = 2.35 + 2πn
  • x = -2.35 + 2πn

Example: cos(x)=0.4

• Use calculator to do arccos(0.4) and get principle value: 1.16 Rad.
• Apply trig identity cos(θ) = cos(-θ) to get the mirror solution: arccos(0.4) = - arcsin(0.4), which'd be: -1.16 Rad.
• With two solutions 1.16 Rad and -1.16 Rad and add periodicity, we'll get two full solutions:
  • x = 1.16 + 2πn
  • x = -1.16 + 2πn

Range of the inverse trig functions

The range differs for each arc-function:

• Arcsin(x)=θ: -90° < θ < 90°, means angle only exists in Q.1 and Q.4
• Arccos(x)=θ: 0° < θ < 180°, means angle only exists in Q.1 and Q.2
• Arctan(x)=θ: -90° < θ < 90°, means angle only exists in Q.1 and Q.4

Tricks:

About the signs like 90 and -90,
just to think about the CLOCKWISE and COUNTER-CLOCKWISE.

Special Value of trig function solving

With special trig values, we really don't need calculator at all, but only to look at the picture of Complete Unit Circle. Or not even that if you can remember it.

Notice:
The Complete Unit Circle only shows counter-clockwise angle measures which means ONLY POSITIVE ANGLES,
so you have to do your own math to get the NEGATIVE ANGLES, aka. clockwise angle measures.

Refer to youtube: Evaluating Inverse Trigonometric Functions, Basic Introduction, Examples & Practice Problems: sin⁻¹(1/2), sin⁻¹(√3/2), [sin⁻¹(-1/2)], sin⁻¹(-√3/2), sin⁻¹(0), sin⁻¹(1), sin⁻¹(-1), cos⁻¹(1/2), cos⁻¹(-√3/2), cos⁻¹(-√2/2), cos⁻¹(0), tan⁻¹(0), tan⁻¹(1), tan⁻¹(-1), tan⁻¹(√3), tan⁻¹(-√3/3), review all.

Example: Solve sin⁻¹(1/2)

• By looking at the unit circle, we know there're multiple arc measures could get a sine value 1/2, which are 30°, 150°, -210°, -330°.
• Filter out all others by the Range [-90°, 90°], we will get 30° is the ONLY-ONE and MUST-ONLY-ONE answer.

Example: Solve sin⁻¹(-1/2)

• Look at the unit circle, we know the answer are 210°, 330°, -30°, -150°
• With filtering by range [-90°, 90°], so -30° is the only answer.

Example: Solve cos⁻¹(√2/2)

Basic trig equations

Once you figure out how to solve the original functions solution, it's so easy with this basic one.
Step-by-step solutions are as below:

• Simplify the equation to sin(θ)=?? form.
• Solve θ with TWO solutions in the form of θ = ?? + 2πn.
• Replace θ with the expression of x and solve the equation for x.

Khan practice.

Example: Solve 20sin(10x) − 10 = 5

• Simplify the equation and get sin(θ) = 3/4
• Solve sin(θ) = 3/4 get solutions for is: θ = 0.85 + 2πn and θ = 2.29 + 2πn
• Replace θ with expression of x, which make 10x = 0.85 + 2πn and 10x = 2.29 + 2πn
• Solve equations for x, get x = 0.085 + 0.2πn and x = 0.229 + 0.2πn

Composition of Inverse trig functions

https://www.youtube.com/watch?v=pWdGu9E5nCE

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Reference Angle

The notion reference angle would be the very FIRST thing to understand before solving any trig equations.

First thing first, it's always the angel with X-axis and Terminal Ray.

What is it for:
In trigonometry, if angles have the same reference angle, then they do have the SAME trig function values.
e.g. sin(x) = sin(π-x),
cos(x) = cos(2π-x).

Math bits notebook.
Math open reference.

Features of a reference angle:

• It's always LESS THAN 90°.
• It's always POSITIVE.
• It's always the angle between the Side and the X-axis.

How to get the reference angle θ:

• Quadrant I: θ
• Quadrant II: π - θ
• Quadrant III: θ - π
• Quadrant IV: 2π - θ

Angles in Standard Position

Regard to the reference angle, the angles in standard position means the ORIGINAL angle.

e.g. There's an angle θ = 330°. So 330° is the angle in standard position, and 30° is its reference angle.

Tricks:

To imagine a Diving Athlete finished a ROLLING before hits water, she rolled may be 2 rounds, aka. 720°, and she made a 45° angle with her body and the surface of water.

Reference Triangles

A reference triangle is formed by "dropping" a PERPENDICULAR line from the terminal ray of a standard position angle to the X-axis. Remember, it must be drawn to the x-axis.

Math is fun.

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Signs of trig functions

Before solving a trig equation, need look at the sign of the equation, AND identify which one or multiple quadrants will the solutions land in.

• Quadrant I: All trig functions are POSITIVE.
• Quadrant II: Only sin(x) is POSITIVE, others are all NEGATIVE.
• Quadrant III: Only tan(x) is POSITIVE, others are all NEGATIVE.
• Quadrant IV: Only cos(x) is POSITIVE, others are all NEGATIVE.

e.g.
sin(x) = 123 have a positive sign, so it will have two solutions: one in Quadrant I, another be Quadrant II.
-cos(2x+3)=234have a negative sign, so it'll have two solutions in Quadrant II and Quadrant III.

Tricks:
In a unit circle, X-axis could be represented by sine, Y-axis by cosine.
So just think how x and y will be positive and negative in each quadrant will give you the answer.
Btw, tangent is equal to sin(x)/cos(x), so it's easy to guess its sign too.

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Unit circle

It's a fairly fundamental tool for solving trigonometric problems.

Refer to CalcWorkshop.

"Unit Circle is nothing more than a circle with a bunch of Special Right Triangles."

Well, these special right triangles help us in connecting everything we’ve learned so far about Reference Angles, Reference Triangles, and Trigonometric Functions, and puts them all together in one nice happy circle and allow us to find angles and lengths quickly.

Special right triangles

30-60-90 Triangle:

45-45-90 Triangle:

Unit Circle with Special Right Triangles:

The Complete Unit Circle

X-axis in the image is Cosine-Axis, and Y-axis is Sine-Axis.

Exact values of the unit circle

How to memorise the Unit Circle

Refer to this PDF: unit-circle-radian-measure.pdf

Actually, we only need to remember the First Quadrant, and just REFLECT it about Y-axis and X-axis and the Origin, then we get all four quadrants of these special values.

Something important to remember:

• Each point on the unit circle, has other THREE images on the unit circle, so in total it's FOUR of them. Two are POSITIVE, other two are NEGATIVE.

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Average in STATS

Average in statistics means bit different than just a arithmetic average.

Khan lecture.

Average: In stats, it means typical or middle, and could be represented by multiple ways:

• Arithmetic mean: Sum numbers and get average.
• Median: Sort numbers and get the MIDDLE one.
• Mode: A number repeats the most times in a dataset.

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Empirical rule (STATS)

It's a rule describing a Normal Distribution.

It's also called the 68-95-99.7% rule, because for a normal distribution:

• ≈68% of the data falls within 111 standard deviation of the mean
• ≈95% of the data falls within 222 standard deviations of the mean
• ≈99.7% of the data falls within 333 standard deviations of the mean

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Permutations & Combinations (PROBABILITY)

Aside from probability, Permutations and Combinations are essential tools for statistics.

They're to solve the problem: how many groups are there of if we choose some from some.

Jump back to previous note: Intro to probability.

HOW MANY groups do we get if we choose a number things from the total things?
e.g., how many groups would there be if we choose 3 people from 9 people?

Permutations and combinations are both to count the total number of groups.
We got TWO types of ways to count:

• Permutation: If the order in the group matters.
• Combination: If the order doesn't matter.

Combinations could be seen as FILTERED permutations, which filtered out all the "duplicates", or "over counted items".

e.g., We got different groups(Permutations) as "123, 132, 231, 213, 312, 321", once we filter out the over counted items,
the combination is just one: 123.

Refer to web article.

Permutations

It's literarily saying the possibilities.

Notice: possibilities ≠ probabilities

e.g., the possibilities of how to arrange three numbers 1,2,3?
It could be: 123, 132, 231, 213, 312, 321, so answer is 6 possible ways.
To count that algebraically, it'd be 3*2*1, answer is 6 possible ways.

How do we do this?

Possible ways to fit in the 1st position are 3, and we got 2 left overs. Then the 2nd place could have 2 possible ways, and we got 1 left over. So the 3rd position could be 1 possible way.

And just to logically think about it, we should MULTIPLY them together to get ALL POSSIBLE WAYS: 3*2*1.

Combinations

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Ellipse and its equation

Ellipse is twisted circle, it got major features and equation from circle.

Ellipse got TWO radiuses (or radii):

• Major radius: the longer one.
• Minor radius: the shorter one.

Standard ellipse equation

a, b is the radius on x-axis and y-axis. Ellipse originally centred at (0, 0).

Foci of ellipse

Since ellipse is part of Conic section, so it has the same identity:

There're TWO points on the Major radius of ellipse, so that any point on this ellipse has the same sum of distance to both points. These two points are called Focuses or Foci.

How to find the Foci

• Major radius on X-axis: f² = a²-b²
• Major radius on Y-axis: f² = b²-a²

Foci identity

With the distance D₁ and D₂, their sum is equal to Double Major Radius:

D₁ + D₂ = 2*r

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Hyperbola and its equation

Refer to Maths is fun.

Hyperbola can be defined as a curve where the distances of any point from:

• A point (the focus)
• A (the directrix) are always in the same ratio.

Equation of Hyperbola

Equation of hyperbola opens on the X-axis:

Equation of hyperbola opens on the Y-axis:

Features:

• Two vertices:
  • (±a, 0) when it opens on x-axis.
  • (0, ±b) when it opens on y-axis.
• Two asymptotes:
  • y = (b/a)x
  • y = −(b/a)x

Foci of hyperbola

Any point P on the hyperbola to both focuses

|D₁ + D₂| = CONSTANT

How to find foci of hyperbola

f² = a² + b²

f is the focal length, which is the distance from the focus to the centre.
a is the distance from the vertex to the centre, when the vertices are on X-axis.
b is the distance from the vertex to the centre, when the vertices are on Y-axis.

Example

Solve:

• Its centre is at origin, so the unknowns are x² & y².
• According to its vertices, the hyperbola opens Up & Down.
• Since it's up&down, the formula is y²/b² - x²/a² = 1.
• And the b² = √(6)² = 6
• According to the foci equation: then it should be 74 = a² + 6, and a² = 68.
• So the equation would be y²/6 - x²/68 = 1

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Complex numbers

Imaginary number

# Assume that
i² = −1
# therefore the imaginary unit is
i = √(-1)

Powers of Imaginary Unit:

The pattern is from i to i⁴, then it repeats.
To evaluate iⁿ, just to get the remainder n % 4 and see which term it is in the pattern. e.g., i¹⁰ is equal to -1 because 10 % 4 = 2 which is 2nd term in the pattern.

Complex Plane

Conjugates

Its mainly usage is for the complex number's division.

Forms of complex number

refer to khan academy article.

• Rectangle form
  This form is good for plotting in the complex plane.
• Polar form
  This form only refers to the Absolute value and Angle,
  which are also called the Modulus and the Argument.
• Exponential form
  Its reasoning is too complicated, so just to remember it.
  This form is very good for complex number's multiplication and division or further operations.
  

Operations of complex numbers

It's easier just see the i as an unknown variable, then to do normal operations.
Refer to maths is fun.

• Addition & Subtraction:
  
• Multiplication:
  
• Division:
  Need to use the notion conjugates
  
• Powers:
  Better to convert it to Exponential form first, and do powers.
  

Convert between Rectangle form & Polar form

FORMS AND TECHNIQUES HERE, ARE RATHER APPLICABLE TO MANY TOPICS THAN MERELY FOR COMPLEX NUMBER. IT'S ACTUALLY FOR ALL TRIGONOMETRY RELATED TOPICS!!

Absolute value of complex number

The absolute value of it literarily means the DISTANCE from the point of complex number to the origin.

How to understand this term?

IT SPEND ME A LOT OF TIME TO UNDERSTAND AT THE FIRST TIME.
In the example 5+6i,
to calculate the absolute value of it |5+6i|, I know it's using the Pythagorean theorem.

But why it ISN'T doing √(5²+6²i²) instead of √(5²+6²)?
It is the most confusing part, and hard to consult the Internet with a clear answer that why do we Take out the i from calculating the distance.

The simple answer is that:
No matter it's i or i², it is either √(-1) or -1, in another word it actually is just a NEGATIVE ONE,
but when we're counting a DISTANCE, it's always a POSITIVE, so we have to CANCEL OUT the -1 from the number. That's the point we could ignore it when doing absolute value.

Angle of complex number

Note:
It only regards to the tangent knowledge of trig functions.
Better to review tangent identities to find all possible solutions of tan⁻¹(θ).

tan(θ) = tan(-π + θ)

Steps:

• Find all solutions for angle θ.
• Plot rectangle form of complex number and figure out the Quadrant.
• Select the right solution at the right quadrant.

Example: Find angle for z = −3 − 6i
Solve:

• Principle solution: θ = arctan(-6/-3) = 1.107 which is at 1st Quadrant.
• Another pair solution would be -π + θ = -2.03 which is at 3rd Quadrant.
• -3-6i should be at the 3rd Quadrant in complex plane, so answer is -2.03.

For some exact values, we also need the tangent values of unit circle:

Convert between Exponential form & Polar form

Example: Find the solution of z⁴ = -625 in Rectangle form and Polar form, which argument in [270°, 360°]
Solve:
It's actually a process of conversion:

• -625 could be represented as -625 +0i in Rectangle form
• Modulus r is √(-625²+0²) which is equal to 625.
• tanθ = 0/-625 = 0, check the tangent unit circle table to know the angle could be 0° or 180°, according to its rectangle form, it's certain to be on the negative X-axis, so it's 180°.
• Add all possible solutions and get the angle is 180° + n*360°.
• Polar form is: 625[cos(180° + n*360°) + i*sin(180° + n*360°)]
• since the complex number is z⁴, so:
  • Modulus = ⁴√625 = 5
  • Argument = (180° + n*360°) / 4 = (45°+n*90)
  • Range of argument is [270°, 360°], so the argument is 315°
  • So the Polar form is: 5(cos315 + i*sin315).
  • Rectangle form is: 5√2/2 - i*5√2/2

Powers of complex numbers

Example

Solve:

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Studies design (Statistics)

Refer to Khan academy course.

Types of studies:

• Sample study: To estimate ONE certain parameter of a population.
• Observational studies: To compare TWO parameters of a population WITHOUT affecting them.
• Experiments: To compare TWO parameters of a population and DOES affecting one and does not affecting the other one.

Samples or surveys

• Problem 1: Qualitative and quantitative data
• Problem 2: Representative samples
• Problem 3: Biased wording in survey questions
• Problem 4: Sampling methods
  • Systematic sampling: 100% members of ALL groups chosen.
  • Stratified sampling: Some members from ALL groups chosen.
  • Random sampling: An adequate number of members chosen, each an equal chance of being in the sample.
  • Cluster sampling: 100% members from SOME groups are chosen.

Observational studies & experiments

• Observational study: Measure or survey members of a sample WITHOUT trying to affect them.
• Controlled experiment: Apply some treatment to one of the groups, while the other group does not receive the treatment.

Notes

• Randomized experiments are designed to suggest causation
• Correlation is WEAKER than causation
• To answer a question about a causal relationship, we need to perform an experiment with a treatment group and a control group.
• While sample study need a part of relative members, Observational study need ALL members.

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Trapezoid

Definition: A trapezoid is a quadrilateral with two parallel sides.

To play with trapezoid, Refer to GeoGebra

Trapezoid

• Scalene trapezoid: No equal sides.
• Isosceles trapezoid: Two legs are equal.
  

Area of trapezoid

Trapezoid inscribed in a circle

• Remember that a trapezoid has to have TWO BASES to be parallel.
• Know that, a quadrilateral CAN be inscribed in a circle or even a semicircle, which means 4 vertices are all on the circle.
• Since it's a trapezoid, and inscribed in a circle, then IT HAS TO BE A ISOSCELES TRAPEZOID. No matter its side crosses the centre or not.

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Partial Fractions

It's a very useful trick even for Calculus problems

▶ Jump forward to the note on application on Partial fractions for Calculus Integration.

Example

Solve:

• Factorize the dominator.
• Write Partial fractions according to the dominator.
• Add up the partial fractions and arrange it in the same form of nominator:
  
• Compare the nominators of both fraction:
  
• Build System of equations to solve out for A & B:
  
• Plug in and write the Partial fractions:
  

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Find roots of polynomials

Example

Solve:

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Direct & Inverse variation

When we say two variables vary directly, it's like this:
y = mx
Their absolute value go the same way up or down.

But they vary inversely when it's like this:
y = m/x
Their absolute value go opposite way that one goes up the other goes down.

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Positional number

Refer to Wikipedia: Positional notation

We often hear that ones place, tenth place, hundreds place or more. Here is the graph helps easily to understand.

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Significant Figures (Significant digits)

The term "significant figures" refers to the digits, in a number or measurement, which show the accuracy of the number.

Refer to wiki: Significant figures
Refer to Khan academy: Intro to significant figures

Rules of Significant figures

Concise Rules:

• All non-zero digits are significant: 1, 2, 3, 4, 5, 6, 7, 8, 9.
• Zeros between non-zero digits are significant: 102, 2005, 50009.
• Leading zeros are NEVER significant: 0.02, 001.887, 0.000515.
• In a number with a decimal point, trailing zeros (those to the right of the last non-zero digit) are significant: 2.02000, 5.400, 57.5400.
• In a number without a decimal point, trailing zeros may or may not be significant. More information through additional graphical symbols or explicit information on errors is needed to clarify the significance of trailing zeros.