Finding values of sine without a calculator

How do I find the sin or inverse sin of 46 degrees (or any degree) without using a calculator?
Thanks.
Jackie

Finding an "inverse sin" doesn't apply to degrees by the way. We take sine of an angle and get a number; inverse sine is taken from a number and gives back an angle.

So is it possible to find values of sine without a calculator? I am sure there are various methods, but these two came to my mind.

1) We can go back to the definition of sine in a right triangle and using a protractor, DRAW a right triangle with that angle. Draw as accurately as you can.



Then from the picture again, we need to measure the t! wo sides: the opposite side and the hypotenuse and then calculate their ratio (paper and pencil). Again, measure as accurately as you can.

Now, as far as the opposite problem, let's say you know that the sine of some angle is 0.86 or some other number (between -1 and 1). Can we find the angle without the calculator?

Draw a right triangle with hypotenuse 1, opposite side 0.86 (or some multiple of those), and measure the angle in degrees.

This method, I feel, is a good one for demonstration and teaching purposes when first learning about sine.

2) Another possibility is to use the Taylor series of sine. Hopefully you don't need to take too many terms from it to get the desirable accuracy. This will include calculations that you'd need to do on paper.

Let's say we use Taylor series in origin and take the first four terms:
!
sin(x) ≈ x - x³/3! + x⁵/5! -! x⁷ /7!

To use this, you need to first change the 46 degrees or whatever to radians. Obviously all the calculations involved will take some time without a calculator... Auch! But an approximative method such as this that only involves the four basic operations is what your calculator probably uses, too.

Here you will find the Taylor series for inverse of sine.

3) Using sine addition formula and a known value.

Sine addition formula says:
sin(a + b) = sin a cos b + cos a sin b.

So... if you're interested in finding, say, sin(46°) and we do already happen to know sin(45°) and cos(45°)... but we need to work in radians to use the formula. So convert 46 and 45 degrees to radians (without a calculator? I'm going to cheat now...) and get 45° is Pi/4 or 0.785398, 46° is about 0.80285.

sin(Pi/4 + 0.01745) = sin(Pi/4) cos(0.01745) + cos(Pi/4) sin(0.01745).

It just so happens that for small values of x (near zero), sin x ≈ x. (You learn that in calculus, I think). So sin(0.01745) is about 0.01745. Cos(0.01745) should be pretty near 1 somewhere.

sin(Pi/4) or sin 45° is 1/√2, and cos(Pi/4) is the same. Plugging those in,

sin(Pi/4 + 0.01745) = 1/√2 * 1 + 1/√2 *0.01745.

I'm going to cheat again and do this with a calculator... to get 0.71945.

Did I get close? Well, sin46° is about 0.7193398. Got three decimals right; that's okay I guess.

Tags: sine, calculator

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Ask a Korean! Wiki -- Korean Language Tutors?

Dear Korean,

The American English Professor Husband to Korean Woman and Father of a Very Cute Half Breed Kid with a Half English Half Korean Name (hereinafter referred to simply as "AEPHKWFVCHBKHEHKN") just read The Korean's blog post about learning languages. AEPHKWFVCHBKHEHKN agrees with the Korean, and gives Kudos to the hard-assed realism. People ask how AEPHKWFVCHBKHEHKN learned to speak good Arabic in just a year, and the only answer is pure hard work. (And AEPHKWFVCHBKHEHKN is a big fan of Pinker's book too.)

AEPHKWFVCHBKHEHKN has a question about language tutors. Can the Korean recommend any good tutors or language programs here in Seoul? Does the Korean himself tutor? AEPHKWFVCHBKHEHKN only has a few "free weeks" this summer, so Yonsei's lauded KSL program doesn't work for him. AEPHKWFVCHBKHEHKN asks forgiveness if this information is already on the Korean's blog.

With obsequious thanks to His Munificence, the Korean,
AEPHKWFVCHBKHEHKN

Dear AEPHKWFVCHBKHEHKN,

The Korean is glad you enjoyed the post. But he never had to learn Korean in Korea via tutoring, so he does not know at this time -- but he is sure some of AAK! readers will be able to help you. Readers, any ideas?

Got a question or a comment for the Korean? Email away at askakorean@gmail.com.

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Strategies to solve simple math equations

What are the strategies for solving simple equations?

I got this question in the mail just today.

I assume the person means LINEAR equations - those where you only have one variable (usually x), and that x is not raised to second or third or any other power, nor is it in the denominator or under square root sign or anything. Just x's multiplied by numbers and numbers by themselves, such as:

2x - 14 = 9x + 5

OR

1/3x - 3 = 2 - 1/2x

OR

2(5x - 4) = 3 + 5(-x + 1)

Here are the strategies for solving these:
* You get rid of paretheses using distributive property
* You may multiply both sides of the equation by the same number
* You may divide both sides of the equation by the same number
* You may add the same number to both sides of the equation
* You may subtract the same number from both sides of the equation

You might think, "W! hich one of those will I use, and in which order?"

That depends. There is no clear cut-n-dried answer.

Whatever you do, you try to transform your equation towards the ultimate goal: where you have x on one side alone. Also whatever you do, your goal is to transform the equation to one that you already know how to solve. It might take several steps.

For example, your first step with these equations, could be to...

1/3x - 3 = 2 - 1/2x	... multiply both sides by 6 to get rid of the fractions
2(5x - 4) = 3 + 5(-x + 1)	...multiply out the parentheses
2x - 14 = 9x + 5	...add 14 to both sides (or subtract 9x)
1/4(2x - 27 + 0.5x) = 2/5(8x + 3)	...multiply by 20 to get rid of the fractions

As with most things, practice makes perfect. ! Check also the websites below:

Tutorial on linear equations has a 4-step strategy for solving linear equations which summarizes it real well.

Algebra 1 Review - Solving Simple Equations - a step-by-step slideshow.

Ask Dr. Math ® - Solving simple linear equations - lots of examples to read here.


Tags: math, mathematics, teaching, algebra

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Dividing decimals

I feel students need to get grounded conceptually in this topic. So many times, all they learn about decimal division are the rules of how to go about decimal division when using long division, and it becomes an "empty" skill - a skill that lacks the conceptual foundation.

So for starters, we can do two different kinds of mental math division problems.

1. Division by a whole number - using mental math
   
   Here it is easy to think, "So much is divided between so many persons".
   
   0.9 ÷ 3 is like "You have nine tenths and you divide it between three people. How much does each one get?" The answer is quite easy; each person "gets" 0.3 or three tenths.
   
   And... remember ALWAYS that you can check division problems by multiplication. Since 3 × 0.3 = 0.9, we know the answer was right.
   
   0.4 ÷ 100 turns out to be an easy problem if you write 0.4 as 0.400:
   0.400 ÷ 100 is like ! "You have 400 thousandths and you divide it between 100 people; how much does each one get?" The answer is of course 4 thousandths, or 0.004. Check: 100 × 0.004 which is 100 × 4/1000 = 400/1000 or 0.400 = 0.4.
   
   Here are some more similar ones:
   
   0.27 ÷ 9
   
   0.505 ÷ 5
   
   0.99 ÷ 11
   ...and you can make more, just think of the multiplication tables.
   



2. Division where the quotient (answer) is a whole number
   
   This time it helps to think, "How many times does the divisor go into the dividend?" In these types of mental math problems, the answer ends up being a whole number. (Of course the teacher has to plan these problems just right.)
   
   For example, 0.4 ÷ 0.2. Ask, "How many times does 0.2 fit into 0.4?" The answer is, 2 times. So 0.4 ÷ 0.2 = 2. Again, we can check it by multiplying: 2 × 0.2 = 0.4.
   
   Other similar division problems to solve ment! ally:
   
   1 ÷ 0.5
   
   3 ÷ 0.5
   0.09 ÷ 0.03
   
   0.9 ÷ 0.1
   
   2 ÷ 0.4
   
   1 ÷ 0.01
   
   ...and so on.
   

This decimal division lesson taken from my Decimals 2 book illustrates these two kinds of mental division problems.

Towards the general case

After the student is familiar with the two special cases above, we can go forward and study decimal division problems in general. Even here, we will divide the problems into two classes, depending on whether the divisor is a whole number or not.

1. The divisor is a whole number.
   
   For example, 3.589 ÷ 4 or 0.1938 ÷ 83. These can simply be solved by long division as they are. Just put the decimal point in the same place in the quotient as where it is in the dividend.
   
   The "stumbling block" may come when the division is not even (this also leads into th! e study of repeating decimals). Generally, you can continue the division indefinitely by tagging zeros to the dividend, such as making 3.589 to be 3.589000. Then when you've continued the division as long as you wish (or as long as the book tells you to do it), cut the decimal off at a desired accuracy and round it.
   
   Typical problem in a textbook would say, "Do 2.494 ÷ 3 and give your answer with 3 decimal digits." For this, you need to do the long division until the fourth decimal digit - so as to be able to round to 3 decimal digits. Since 2.494 does not have four decimal digits, you tag a zero to it to make it have so (2.4940).
   
   Fortunately, this process is not generally difficult. It's the second case that's more of a problem.
   
   



2. The divisor is not a whole number.
   
   Here, we do something quite special be! fore dividing, and turn the problem into one where the divisor! is a whole number. Then, the actual division is done like explained above.
   
   I say this is special, because this special thing that we do is based on a very important general principle of arithmetic:
   
   If you multiply both the dividend and the divisor by some same number, the quotient won't change.
   
   Let's see it in action with some easy numbers:
   
   1000 ÷ 200 = 5
   
   100 ÷ 20 = 5
   
   10 ÷ 2 = 5
   
   Each time both the dividend and the divisor change by a factor of ten, but the quotient does not change.
   
   We can also try it using a factor of 3 (or any other number):
   
   8 ÷ 2 = 4
   24 ÷ 6 = 4
   72 ÷ 18 = 4
   
   Let's try one more time, with a factor of 2:
   
   30 ÷ 6 = 5
   
   15 ÷ 3 = 5
   
   7.5 ÷ 1.5 = 5
   
   3.75 ÷ 0.75 = 5
   
   H hopefully by now you have convinced the student(s) of this pri! nciple. Now we can apply it to those pesky decimal division problems.
   
   
   
   
   This image shows how the decimal division problem 0.644 ÷ 0.023 can be changed into another problem, with a whole number divisor, and with the same answer.
   
   In each step, we multiply both the dividend and the divisor by 10. This, of course, is the same process as moving the decimal point.
   
   Many textbooks only show the student the "trick" of moving the decimal point... but don't show him what that idea is based on.
   
   An example
   
   To solve 13.29 ÷ 5.19, we need to first change the problem so that the divisor 5.19 is a whole number. We multiply both the dividend and the divisor by 10 as many times as needful to accomplish that:
   
   13.29 ÷ 5.19!
   = 132.9 ÷ 51.9
   = 1329 ÷ 519, and now off ! you go t o do long division... I'm not saying it's the easiest long division problem in the world, since the divisor is 519. Let's try an easier one.
   
   
   2,916 ÷ 0.02
   = 29,160 ÷ 0.2
   = 291,600 ÷ 2 and now you can do the long division.
   
   Of course, in reality you can also multiply by 100 instead of taking two steps of multiplying by 10. But students can start out by multiplying by 10 as many times as needed.
   

Please also see the lesson on dividing decimals by decimals, from my Math Mammoth Decimals 2 book.

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The Properties of Real Numbers

Now that we have an alphabet, let's establish the rules of grammar for the language of mathematics. All of these rules work for any real numbers, so we'll use our new variables to express numbers in general. You can use the rules for substituting for variables with any of these rules, just be sure to use different numbers for different variables.

There are several kinds of rules. There are axioms which are so basic of a rule they are simply taken for granted. There are laws that are the consequence of axioms and are true everywhere all the time. And there are theorems which are rules that have been accepted as true because they have been built from axioms and laws and other theorems. Some of them we've already seen.



































!

Closure Property
Adding any two real numbers results in another real number.
Multiplying any two real numbers results in another real number.
Transiti! ve Property
If two numbers are equal to the same number, then they are equal to each other.	If a=c and b=c then a=b
Equality Axioms
If two numbers are equal and you add the same number to each the resulting numbers are equal.	If a=b then a+c=b+c.
If two numbers are equal and you mutliply each by the same number the resulting numbers are equal.	If a=b then ac=bc.
Commutative Property
When adding real numbers, the order doesn't matter.	a+b=b+a
When multiplying real numbers, the order doesn't matter.	ab=ba
Associative Property
When adding real numbers, the grouping doesn't matter.	a+(b+c)=(a+b)+c
When multiplying real numbers, the grouping doesn't matter.	a(bc)=(ab)c
Distri! butive Property
Multiplying a sum of re! al numbe rs by a real number is the same as multiplying each real number of the sum first then adding.	a(b+c)=ab+ac
This is a powerful property. It allows us to pull a common factor out of the parts of an equation and deal with it separately.
Identities
Adding zero to a real number doesn't change the number.	a+0=a
Multiplying a real number by 1 doesn't change the number.	a×1=a
Inverses
Adding the additive inverse of a real number to itself results in the identity.	a+(-a)=0=(-a)+a
Multiplying the reciprocal of a real number to itself results in the identity.	a(1/a)=1 if a≠0
Zero Principle
Multplying by zero results in zero.	a×0=0
If the product of two real numbers is zero, then one of them must h! ave been zero.	If ab=0 then a=0 or b=0
Trichotomy Property
If two real numbers are compared, the second number can have only one of three outcomes, either greater than, less than or equal to.	a<b, a>b or a=b
Definition a^-1	a^-1=1/a
Properties of Negatives
Multiplying a real number by -1 changes the number's sign.	(-1)a=-a
	-(-a)=a
For any negative product of two real numbers, the negative sign can be assigned to either one the numbers.	(-a)b=-ab=a(-b)
The product of two negative real numbers is equal to the product of the additive inverses of the two numbers.	(-a)(-b)=ab
Negating the difference of two real numbers changes to signs of each number.	-(a-b)=b-a
Properties of Quotients
For any two rational numbers, the first rat! ional nu mber is equal to the second rational number if the product of the numerator of the first and the denominator of the second is equal to the product of the numerator of the second and the denominator of the first	a/b=c/d if and only if ad=bc
Mutliplying a rational number by a rational number equal to 1 changes the appearance of the rational number but not its value.	a/b=ad/bd
a/b+c/b=(a+c)/b
a/b+c/d=(ad+bc)/bd
a/b × c/d = ac/bd
a/b ÷ c/d = a/b × d/c = ad/bc
Definition
The principle square root of a number √a is the nonnegative real number b such that b squared is equal to a.	b²=a
Laws of Exponents
Definition
The absolute value of a real number is the magnitude of the number without a s! ign.	|a|=a, |-a|=a
Absolute Value Properties
The absolute value of a real number has the following properties
Positive Definite
For all real numbers, the absolute value of the number a is greater than or equal to 0 with |a|=0 only if a=0
Symmetric
For any two real numbers, |a-b|=|b-a|
Triangle Inequalities
For any two real numbers, the absolute value of the sum of the numbers is less than or equal to the sum of the absolute values of the numbers.	|a+b|≤|a|+|b|
For any two real numbers, the absolute value of the difference of the numbers is greater than or equal to the difference of the absolute values of the numbers.	|a-b|≥|a|-|b|
For any two real numbers, the absolute value of the difference of the absolute va! lues of the numbers is less than or equal to the absolute valu! e of the difference of the numbers.	||a|-|b||≤|a-b|

properties of real numbers

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Yearly Plan – Additional Mathematics Form 4 (2009)

Yearly Plan – Additional Mathematics Form 4 (2009)

 

 

 

Involve algebraic functions only.

 

 

 

Images of composite functions include a range of values. (Limit to linear composite functions).

Define composite functions

 

 

 

 

 

 

 

 

 

!
 

 

 

 

Week No	Learning Objectives Pupils will be taught to.....	Learning Outcomes Pupils will be able to…	No of Periods< /span>	Suggested Teaching & Learning activities/Learning Skills/Values	Points to Note
Topic/Learning Area Al : FUNction --- 3 weeks First Term
2   5-9/1/09	1. Understand the concept of relations.	1.1 Represent relations using arrow diagrams ordered pairs graphs Identify domain, co domain, object, image and range of a relation. 1.3 Classify a relation shown on a mapped diagram as: one to one, many to one, one to many! or many to many relation.	1               1	Use pictures, role-play and computer software to introduce the concept of relations.     Skill : Interpretation, observe connection between domain, co domain, object, image and range of a relation.	Discuss the idea of set and introduce set notation.
	2.    Understand the concept of functions.	2.1 Recognise functions as a special relation..   2.2 Express functions using function notation.   2.3 Determine domain, obje! ct, image and range of a function. 2.4 Determine the image of a function given the object and vice versa.	1         1	Give examples of finding images given the object and vice versa. Given f : x ® 4x – x2. Find image of 5. Given function h : x ® 3x – 12. Find object with image = 0.   Use graphing calculators and computer software to explore the image of functions.	Represent functions using arrow diagrams, ordered pairs or graphs, e.g.   "" is read as "function f maps x to 2x". ""is read as "2x is the image of x under the function f". Include examples of functions that are not mathematically based. Examples of functions include algebraic (linear and quadratic), trigonometric and absolute value. Define and sketch absolute value functions.
3   12-17/1/09	3. Understand the concept of composite functions.	3.1 Determine composition of two functions.     3.2 Determine the image of composite f! unctions given the object and vice versa 60; 3.3 Determine one of the functions in a given composite function given the other related function.	1         1         2	Use arrow diagrams or algebraic method to determine composite functions. Give examples of finding images given the object and vice versa for composite functions   For example : Given f : x ® 3x – 4. Find ff(2), range of value of x if ff(x) > 8.   Give examples for finding a function when the composite function is given and one other function is also given.   Example : Given f : x® 2x – 1. find function g if The composite function fg is given as fg : x ® 7 – 6x composite function gf is given as gf : x ® 5/2x.
4   19-23/1/09	4. Understand the concept of inverse functions.	4.1 Find the object by inverse mapping given its image and function.   4.2 Determine inverse functions using algebra.   4.3 Determine and state the condition for existence of an inverse function Additional Exercises	1   1       1 1	Use sketches of graphs to show the relationship between a function and its inverse.   Exa mples : Given f: x, find	Limit to algebraic functions. Exclude inverse of composite functions.         ! Emphasise that the inverse of a function is not necessarily a function.
5   26-30/1/09		PUBLIC HOLIDAY (CHINESE NEW YEAR)!
Topic A2 : Quadratic Equations ---3 weeks
6   2-6/2/09	1. Understand the concept of quadratic equations and their roots.	Recognise a quadratic equation and express it in general form.     1. 2 Determine whether a given value is the root of a quadratic equation by substitution; inspection.   1.3 Determine roots of quadratic equations by trial and i! mprovement method.	1             1             !	Use graphing calculators or computer software such as the Geometer's Sketchpad and spreadsheet to explore the concept of quadratic equations           Values : Logical thinking Skills : seeing connection, using trial and error methods	Questions for 1..2(b) are given in the form of ; a and b are numerical values.
7   9-14/2/09	2. Understand the concept of quadratic equations.	2.1 Determine the roots of a quadratic equation by factorisation; completing the square c) using the formula.     2.2 Form a quadratic equation from given roots. !	1   1       2	If x = p and x = q are the roots, then the quadratic equation is , that is . Involve the use of: and where α and β are roots of the quadratic equation   Skills : Mental process, trial and error	Discuss when , hence or . Include cases when p = q.   Derivation of formula for 2.1c is not required.
8   16-20/2/09	3. Understand and use the conditions for quadratic equations to have a) two different roots; b) two equal roots; c) no roots.    a)dua punca berbeza;	3.1 Determine types of roots of quadratic equations from the value of .   3.2 Solve problems involving in quadratic equations to: a) find an unknown value; b) derive a relation.   Additional Exercises	2       2           2	Giving quadratic equations with the following conditions : , and ask pupils to find out the type of ro! ots the equation has in each case.   Values: Making conclusion, connection and comparison	Explain that "no roots" means "no real roots".
Topic A3 : Quadratics Functions---3 weeks			!
9   23-27/2/09	1. Understand the concept of quadratic functions and their graphs.	1.1 Recognise quadratic functions	1	1) Use graphing calculators or computer software such as Geometer's Sketchpad to explore the graphs of quadratic functions. f(x) = ax2 + bx + c f(x) = ax2 + bx f(x) = ax2 + c * pedagogy : Constructivism Skills : making comparison & making conclusion
		1.2 Plot quadratic function graphs:      a)based on given tabulated values; b) by tabulating values based on given functions.	2	1) Use examples of everyday situations to introduce graphs of quadratic functions.   Contextual learning
		1.3 Recognise shapes of graphs of quadratic functions.	1		Discuss the form of graph if a > 0 and a < 0 for   Explain the term parabola.
10   2-6/3/09		1.4 Relate the position of quadratic function graphs with types of roots for .	2	Recall the type of roots if : b2 – 4ac > 0 b2 – 4ac < 0 b2 – 4ac = 0	Relate the type of roots with the position of the graphs.
	2. Find the maximum and minimum values of quadratic functions.	2.1 Determine the maximum or minimum value of a quadratic function by completing the square.	2	Use graphin! g calculators or dynamic geometry software such as the Geometer's Sketchpad to explore the graphs of quadratic functions   Skills : mental process , interpretation	Students be reminded of the steps involved in completing square and how to deduce maximum or minimum value from the function and also the corresponding values of x.
11   9-13/3! /09	3. Sketch graphs of quadratic functions.	3.1 Sketch quadratic function graphs by determining the maximum or minimum point and two other points.	2	Use graphing calculators or dynamic geometry software such as the Geometer's Sketchpad to reinforce the understanding of graphs of quadratic functions. Steps to sketch quadratic graphs: a) Determining the form"È" or "Ç" b) finding maximum or minimum point and axis of symmetry. c) finding the intercept with x-axis and y-axis. d) plot all points e) write the equation of the axis of symmetry	Emphasise the marking of maximum or minimum point and two other points on the graphs drawn or by finding the axis of symmetry and the intersection with the y-axis. Determine other points by finding the intersection with the x-axis (if it exists).
	4. Understand and use the concept of quadratic inequalities.	4.1 Determine the ranges of values of x that satisfies quadratic inequalities.	2	Use graphing calculators or dynamic geometry software such as the Geometer's Sketchpad to explore the concept of quadratic inequalities.	Emphasise on sketching graphs and use of number lines when necessary.
12   16-20/3/09		SCHOOL HOLIDAY
Topic A4: Simultaneous equations---2 weeks
13   23-27/3/09	1. Solve simultaneous equations in two unknowns: one linear equation and one non-linear equation.	1.1 Solve simultaneous equat! ions using the substitution method.	4	Use graphing calculators or dynamic geometry software such as the Geometer's Sketchpad to explore the concept of simultaneous equations. Value: systematic Skills: interpretation of mathematical problem	Limit non-linear equations up to second degree only.
14   30-3/4/09		1.2S olve simultaneous equations involving real-life situations.       Additional Exercises	2           2	Use examples in real-life situations such as area, perimeter and others.   Pedagogy: Contextual Learning Values : Connection between mathematics and other subjects
Topic G1. Coordinate Geometry---5 weeks
! 15   6-10/4/09	1. Find distance between two points.	Find the distance between two points, using formula	1	Skill : Use of formula	Use the Pythagoras' Theorem to find the formula for distance between two points.
	2.    Understand the concept of division of line segments	2.1Find the midpoint of two given points.   2.2Find the coordinates of a point that divides a line according to a given ratio m : n.	1   2	Skill : Use of formula Value : Accurate & neat work	Limit to cases where m and n are positive. Derivation of the formula is not required.
16   13-17/4/09	3    Find areas of polygons.	3.1 Find the area of a triangle based on the area of specific geometrical shapes.   3.2 Find the ar Subscribe to: Posts (Atom) Followers Blog Archive ▼  2010 (34) ▼  August (34) Finding values of sine without a calculator Ask a Korean! Wiki -- Korean Language Tutors? Car And Truck Accessories Free math tutoring online Strategies to solve simple math equations HTC HD2 Apps: Panoramic Calc Pro v2.9.0 Dividing decimals Orimath Quadratic Equation and Function Solver The Properties of Real Numbers Math online tutoring Yearly Plan – Additional Mathematics Form 4 (2009) Finance Problem 1 Factoring the time Finding values of sine without a calculator A factoring trick GFXnew -Yor Best GFX Place Pre Algebra Best Websites Solving Radical Equations The Gauntlet How to find gcd and lcm in Java Introducing the Integral Calculus in Edwards & Penney Minimum distance between two parallel lines - Part II Work Problem 5 NXT Tribot ADOBE ACROBAT PRO EXTENDED 9.0 Dynamical Systems - 8/4/10 Getting to the Core of Our Math Problem Trigonometry - Tangent, SOHCAHTOA Trigonometry - Secant, Cosecant, Cotangent Statistics & Probability Online Assignment Help at... Summer Packet for Students Entering 6th Grade Equivalent Resistance Circuit Examples Sovereign Grace Church (SGC) Co-op Saxon Advanced ... typeonline - free online touch typing course in fi... About Me mathqa48 View my complete profile Simple theme. Powered by Blogger.