# The Basics of High School Math 5 replies

Kr1nge

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7th June 2004

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#1 15 years ago

I hope this helps any of you taking summer school math.

----------------------------------------------------------- PART I — GEOMETRY

Section 1 – Linear Geometry

Regarding Two points (x1,y1) and (x2,y2) Slope: (y2-y1)/(x2-x1) Distance: sqrt((x2-x1)²+(y2-y1)²) Midpoint: ( (x1+x2)/2 , (y1+y2)/2 )

Point Line Distance Between point (x0,y0) and line ax+by+c=0, the perpendicular distance is Section 2 – Planar Geometry

s = side; r = radius; d = diameter; b = base; h = height; (trapezoid only) b1, b2 are two parallel sides; d1, d2 are diagonals; (triangle only) a, b, c, are sides

Areas Square: s² Circle: pi r² Triangle: bh/2 Rectangle: bh Parallelogram: bh Trapezoid: 1/2 (b1 + b2)h Rhombus: 1/2 d1 d2

Perimeters Square: 4s Circle (circumference): 2 pi r = pi d Triangle: a + b + c Rectangle: 2b + 2h

Circles Length of arc: (angle/360)(2 pi r) Area of sector: (angle/360)(pi r²)

Shoelace algorithm for determining the area of a polygon in the Cartesian plane Too lazy to type up; visit IMSA Math Journal's page on it at http://www.imsa.edu/edu/math/journa...es/Shoelace.pdf .

Misc Diagonal of a square: s sqrt(2) Diagonal of a cube: s sqrt(3) Diagonal of a rectangular solid: sqrt(x²+y²+z²) Number of diagonals for a convex polygon with n sides: n(n-3)/2 Sum of interior angles of regular polygon with n sides: 180(n-2) Heron's formula: Area of a triangle = sqrt(s(s-a)(s-b)(s-c)) where s is semiperimeter

Section 3 – 3 Dimensional (spatial) geometry

s = side; l = length; w = width; h = height; r = radius

Volumes Cube: s³ Rectangular Prism: lwh Cylinder: pi r² h Cone: 1/3 pi r² h Sphere: 4/3 pi r³ Pyramid (bottom can have any number of sides): 1/3 (area of base) h

Surface areas Cube: 6s² Rectangular Prism: 2lw + 2lh + 2wh Cylinder: Lateral: 2 pi r h; Total: 2 pi r² + 2 pi r h Sphere: 4 pi r²

----------------------------------------------------------- PART II — TRIGONOMETRY

Section 1 – Basics (what you learn in school)

Definitions sin A = opposite/hypotenuse cos A = adjacent/hypotenuse tan A = opposite/adjacent csc A = hypotenuse/opposite sec A = hypotenuse/adjacent cot A = adjacent/opposite

Odd/Even functions sin(–A) = – sin A cos(–A) = cos A tan(–A) = – tan A

Pythagorean Identities sin²A + cos²A = 1 tan²A + 1 = sec²A 1 + cot²A = csc²A

Sum/Difference 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 2A = 2 sin A cos A cos 2A = cos²A – sin²A = 1 – 2 sin²A = 2 cos²A – 1 tan 2A = (2 tan A)/(1 – tan²A)

Half Angle Identities Section 2 – Advanced (for math competitions)

Triple Angle Identities sin 3A = 3 sin A – 4 sin³A cos 3A = 4 cos³A – 3 cos A tan 3A = (tan A (tan²A – 3))/(3 tan²A – 1)

Sum to Product Identities Product to Sum Identities ----------------------------------------------------------- PART III — NUMBERS

Section 1 – Rules, Means, Sequences

Divisibility Rules 2: last digit is 0, 2, 4, 6, or 8 3: sum of digits is divisible by 3 4: number made by last 2 digits is divisible by 4 5: last digit is 0 or 5 6: satisfies both divisibility by 2 AND 3 7: take the last digit, double it, and subtract it from the rest of the number. Test if that's divisible by 7. Repeat if needed. 8: number made by last 3 digits is divisible by 8 9: sum of digits is divisible by 9 10: last digit is 0 11: (sum of digits in odd places) - (sum of digits in even places) is divisible by 11

Means Geometric mean of a, b: sqrt(ab) Geometric mean of a, b, c: (abc)^(1/3) Arithmetic mean of a, b: (a+b)/2 Arithmetic mean of a, b, c: (a+b+c)/2 Harmonic mean of a, b: 2/(1/a+1/b) Harmonic mean of a, b, c: 3/(1/a+1/b+1/c)

Sequences Arithmetic Sequence: a = first term, d = common difference, an = nth term nth term: a + (n–1)d Sum of first n terms: n (a + an) / 2 Geometric sequence: a = first term, r = common ratio nth term: a r^(n–1) Sum of first n terms: (a(1 – r^n)) / (1 – r)

Special Sums of Sequences Sum of first n odd natural numbers = n² Sum of first n even natural numbers = n²+n Sum of first n integers = n(n+1)/2

Section 2 – Tricks

Squaring a number with a units digit of 5[i] (n5)² = n(n+1) 2 5 where n is block of integers before 5 i.e. (125)²: 12(12+1) = 156 So it's 15625

[i]Factors Number of factors of a number: Find prime factorization, add 1 to each exponent, then multiply those numbers. i.e. 24: 24=2^3 * 3 So (3+1)(1+1)=8, so it has 8 factors.

Sum of factors of a number: Find prime factorization. Then do as follows: 24=2^3 * 3 Sum of factors = (2^0+2^1+2^2+2^3)(3^0+3^1)=60.

Section 3 – Brute force memorization for math contests ^_^

Primes from 1 to 100: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

Perfect Squares from 1² to 25² Perfect Cubes from 1³ to 10³ Not listed for saving of space. But you still need to know them.

Approximate values of Square Roots sqrt(1)=1 sqrt(2)=1.414 sqrt(3)=1.732 sqrt(4)=2 sqrt(5)=2.236 sqrt(6)=2.449 sqrt(7)=2.646 sqrt(8)=2.828 sqrt(9)=3 sqrt(10)=3.162

No! I'm Spamacus!

426,537 XP

17th June 2003

39,480 Posts

#2 15 years ago

Funny how they never talk about the relationship between the fibonochi sequence and the golden ratio. I found out what it was in sixth grade, I was just fooling around with the calculator during lunch because i was bred and found out something that probably everyone else knew. Oh well, I was still excited. :)

As for high school math, it's too easy, at least for me.

Anyways, this is the progression of math of the decades:

Last week I purchased a burger at Burger King for \$1.58. The counter girl took my \$2 and I was digging for my change when I pulled 8 cents from my pocket and gave it to her. She stood there, holding the nickel and 3 pennies, while looking at the screen on her register. I sensed her discomfort and tried to tell her to just give me two quarters, but she hailed the manager for help. While he tried to explain the transaction to her, she stood there and cried. Why do I tell you this?

Teaching Math In 1950: A logger sells a truckload of lumber for \$100. His cost of production is 4/5 of the price. What is his profit?

Teaching Math In 1960: A logger sells a truckload of lumber for \$100. His cost of production is 4/5 of the price, or \$80. What is his profit?

Teaching Math In 1970: A logger exchanges a set "L" of lumber for a set of "M" of money. The cardinality of set "M" is 100. Each element is worth one dollar. Make 100 dots representing the elements of the set "M." The set "C," the cost of production, contains 20 fewer points than set "M." Represent the set "C" as a subset of set "M." Answer this question: What is the cardinality of the set "P" of profits?

Teaching Math In 1980: A logger sells a truckload of lumber for \$100. His cost of production is \$80 and his profit is \$20. Your assignment: Underline the number 20.

Teaching Math In 1990: By cutting down beautiful forest trees, the logger makes \$20. What do you think of this way of making a living? Topic for class participation after answering the question: How did the forest birds and squirrels feel as the logger cut down the trees. (There are no wrong answers)

Teaching Math In 2000: A logger sells a truckload of lumber for \$100. His cost of production is \$120. How does Arthur Anderson determine that his profit margin is \$60?

Teaching Math In 2005: El hachero vende un camion carga por \$100. La cuesta de production es............."

Kr1nge

GFX Artist / Tech Support Guru

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7th June 2004

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#3 15 years ago
Ensign Riles Teaching Math In 2005: El hachero vende un camion carga por \$100. La cuesta de production es............."

hah! so true....

knipple

I'm too cool to Post

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30th September 2003

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#4 15 years ago

ok then

I love ya'all and everything, but this post makes me want to kill myself.

I took 2 math courses in college, Calc 1 and 2, and I am done!

No! I'm Spamacus!

426,537 XP

17th June 2003

39,480 Posts

#5 15 years ago

Well, actually in my class my homework frequently catches on fire. The idiot who sits next to me brings a lighter with him and likes to play with it. Luckily the teacher knows me really well. My average is currently 101.3% and I will need a -66% on the final to be in danger of getting a D.

I have found New York to be behind many other states in when certain curriculum is taught. Right now in ninth grade in New York I am doing the same thing as I was doing in sixth grade math in Oregon.

Kr1nge

GFX Artist / Tech Support Guru

50 XP

7th June 2004

35 Posts