YOU are better than YOU think. Show yourself how:
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Learn to read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence in
work |
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Logic chapters 1 to 5 re- appear not in sequence, as is or longer, in Volume 1A, Pattern Based Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic mastery makes the hard, easier. Logic mastery leads to better, stronger and richer comprehension. Logic mastery improves reading and writing. Logic mastery ease learning difficulties. Logic mastery gives a headstart. In sum, logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving liinear Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills and concepts are developed. Making the hard easier and clearer will allow earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
More Algebra Hints
8. Trigonometry and Complex Numbers
A. Trigonometry
The simplest way to introduce trigonometric functions (functions on your calculator) is to begin with their unit circle definitions, and then specialize to their right triangle computation with the help of similarity assumptions about triangles, right or scalene. Several steps follow for reading in or besides your trig course.
Step 1.
Draw a unit circle
Your unit of measurement may be one centimeter, one meter, one kilometer, one inch, one foot, one yard, one mile or any other unit. Choose one, or draw a circle and declare its radius to be your unit length.
Exercise for Later: How does similarity assumptions for right triangles imply the results, here the definition of trig functions below, is independent of the choose of unit length?
Step 2.
Let q be an angle. Locate the head of the vector with angle q and length 1 on the unit circle.
Step 3.
The head will have coordinates (a units, b units)
on circle of radius 1 unit.
Put cos(q) =a and sin (q) =b. This defines both sine and cosine for all values of the angle q.
Further trig functions may be defined as follows.
when the divisors are nonzero.
The case where q is between 0 and 90 degrees is considered next.
Step 4 (Right Triangle Trigonometry)
circle of radius 1
unit.
Assume q is between 0 and 90 degrees. Then
For angles between 0 and 90 degrees, similarity of right triangles implies the ratios
if you replace the unit circle right triangle by a similar right triangle.
The latter formulas for may be used to compute 
with any right triangle where sides are labeled opposite and adjacent for an
angle 
The further
trig functions may be defined as follows.
when the divisors are nonzero.
Exercise: Express these further trig functions as ratios of the sides opposite, adjacent and/or hypotenuse of the above right triangle.
A trig course will explain the following in more detail.
Trig functions link the ratio of two sides of a right triangle to cosines, sines and tangents of an angle. Knowledge of two sides in right triangle gives knowledge of the third by means of Pythagorean theorem, and of the values of the trig functions for the angles in the triangle. Computation of unknown side lengths, unknown hypotenuse lengths and unknown angles is useful in land measurement (geo - metry) and also in navigation.
From one-to-one properties of trig functions for angles between 0 and 90 degrees or ½p, one can define (say how to compute) inverse trig functions (more functions on your calculator) to compute the angles from the ratio of sides. Computation with inverse trig functions allows one to obtain polar coordinates for vectors or complex numbers from coordinates, real and imaginary parts, or the length of the adjacent and opposite sides of a right triangle determined by the coordinates. Again, this removes the need to measure the lengths and angles for points with rectangular coordinates [a, b].
Calculation
One may define trig functions by saying how to compute them in principle as above, but then one computes or approximates them in practice from tables and slide rules (old fashioned approach) or using calculators (the new approach). Unfortunately in this practice, the tables, slide rules or calculation devices are black boxes which provide results, but whose derivation or justification is not commonly known. This departs from the principle of understanding the computations one does, but the numbers computed by these black boxes can be checked in simple cases. When calculators first arrived, some used faulty or suboptimal methods (algorithms) to compute.
B. Calculus
Geometric and Algebraic previews of calculus may help senior high school studies in and before calculus.
Calculus in the first instance is the subject of slope related computations, their reversal and interpretations. Calculus is the first course in which the algebraic way of writing and reasoning is required at full strength in several different ways.
C. Trigonometry and Complex Numbers
If z = (r,q) in polar coordinates, then z = a + i b =[a,b] = [r cos(q), r sin(q)] in rectangular coordinates. So the ability to compute cosines and sines avoids the need to measure the rectangular coordinates after a diagram after locating the point z from its polar coordinates.
A quick way, or the quickest way, to understand and fully explain the algebraic properties of trig functions is online is to start a trig courses after this treatment of complex complex numbers
Logs, Powers and Exponentials of Complex Numbers
Preview of Electrical Engineering Mathematics
This last section defines (states formulas for) the exponential, logarithms and powers of complex numbers x+iy etc. If you are a science and engineering student you will eventually meet these functions and see their properties. This chapter gives a list of functions which you should expect to meet and understand in the first two years of your university studies. (The further discussion of these functions is left to a second or third course on calculus. From time to time, you should refer to the definitions given below to see how many on this list remain to be seen in your courses.)
The exponential of a complex number x+iy is given by
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Further, if x+iy = r cos(q) +isin(q) ¹ 0 with - = -180° < q £ 180° = p then the principal value of the natural logarithm
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Note from the above definitions and algebraic properties of ln(x) and exp(x) for real numbers x,
- fundamental properties of exponentials: exp(z1+z2) = exp(z1)exp(z2)
- fundamental property of logarithms: ln(z1z2) = ln(z1)+ln(z2)+i2pn for some integer n Î {0,±1},
- first inverse property: exp(ln(z)) = z if z ¹ 0,
- second inverse property: ln(exp(z))-z = 2npi = ni 360 degrees for some integer n
Quick Definitions
- powers defined: zx+iy = exp((x+iy)ln(z)) for z ¹ 0,
- the definition of a logarithm to the complex base a+ib:
loga+ib(z) = ln(z)
ln(a+ib) - the hyperbolic cosine of the complex number x+iy defined:
cosh(x+iy) = exp(x+iy)+exp(-x-iy)
2What do you get if y = 0?
What do you get if x = 0?
Calculators give cosh(x) - The hyperbolic sine of the complex number x+iy defined:
sinh(x+iy) = exp(x+iy)-exp(-x-iy)
2What do you get if y = 0?
What do you get if x = 0?
Calculators give sinh(x)
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www.whyslopes.com
Algebra, Odds & Ends,
1. Hints for Exams 2A. Exact Arithmetic 2B. Fractions Briefly 3. What is a Variable? 4.. Square Roots 5. Straight Lines 6. Problem Solving Methods 7. Trig and Complex No. 8. Complex Applet 9. History of No.s 10. ln(x) and exp(x) 13. Rename the > Sign 14. Problems: Quadratics 15. Problems: Algebra Test 16. Problems: Linear Eqns I 17. Problems: Linear Eqns II 18. Problem Solving Hints 19. Functions & Sets 20. Independent Variables 21. Why Logic 22. Why Math 23. The 15 Times Table 24. The 20 Times Table 25. Algebra Formulas 26. On Learning Maths 27. Maths in Biology 28. Navigation +Time 29 Quibble-What is Algebra 30. Logic in Maths
Odd and Ends, Essays
Constant Retirement Rate Road Safety 3 Strikes Law in California. Math HOW-TOs 9 Steps in Maths
Study With Others: twiddla.com has developed a collaborative whiteboard with audio & text chat that allows students, tutors & teachers to explore & scribble on blank pages and copies of webpages together, If scribbling is awkward with one browser, try another.
In Volume 2, Three Skills for Algebra, Chapters 8 to 14 and postscript What is a Variable point to a greater & clear use of words in algebra. Chapter 14 introduces a 4th skill for algebra, an elaboration of the third: - The direct and indirect use of formulas, numerically and algebraically, is unifying theme that should be mentioned aloud, with words, in each and every use of formula.