YOU are better than YOU think. Show yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work Read notes and textbooks like a lawyer, so that no nuance, no subtlety and no clause escapes your attention. |
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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.
From Degrees to Radians and Back
Arc length and central angles
For a central angle q determines an arc of a circle. The length s of the arc is proportional to the measure of the angle in say degrees. That is s = kq
Physically, this proportionality means if the angle q is double or tripled, so is the arclength s. If you graphed s versus q where 0° £ q £ 360° , you would get a straight line segment with slope m = k.
The assumption that the circumference of the circle is 2pr
implies 2pr = k·360°
where r is the radius. This in turn gives the value of the
proportionality constant k
|
| s | = kq = | pr
180° |
q |
There-in lies we can use to compute arclength s from a knowledge of the radius r and the angle q . Given any two of the three quantities, the third can be computed,.
Similarity of Sectors of Concentric Circles
Outer arc length s2 is proportion to inner arc
length s1
For a pair of concentric circles of radii r1 and r2 respectively, an central angle q determines an arc of length s1 and s2 in the another. But for both the formula
| s
r |
= | p
180° |
q |
applies. Therefore the number of radii in the arcs of each circle are equal.
| s1
r1 |
= | s2
r2 |
= | p
180° |
q |
The foregoing implies the two sector in concentric circles are similar when their central angles are equal. The ratio of radii gives the proportionality constant.
The foregoing can be extended to sectors of non-concentric circles: Exercise: show: Two sectors are similar when and only when their central angles are equal.
How radii are there in an arc subtended by an angle q?
The formula
| s | = kq = | pr
180° |
q |
implies the number of radii in the arclength s is given by the ratio
|
So the number of radii s/r in the arc is proportional to the angle q with and the proportionality constant is
| p
180° |
Since this proportionality constant is nonzero, we may go back and forth between the number of radii in an arc and the degree measure of the central angle.
Can we compute the central angle from the ratio s/r ?
The equality
|
implies
|
| s
r |
determines the angle q, and vice-versa. So specifying one, specifies the other. We will call the ratio s/r, the radian measure of the angle as it gives the number of radii in an arc determine by a central angle , a number that is independent of the circle radius r > 0 to the similarity of concentric sectors with a common central angle.
Number of Radians to and
from Number of Degrees
The real number
| a | = | s --- r |
equals the number of times the radius r goes into the arc length s determined by a central angle q. This ratio, the real number
| a | = | s --- r |
is uniquely determine by the central angle measure in degrees, and vice-versa. So the central angle may be specified in two different ways, with degrees or with number a of radii in the arc subtended by the central angle.
We say the angle measure is a radians when and only when the the number of radii in radii in the arc subtended by the central angle is given by the number a.
We say the angle measure is N degrees when and only when the the number of degrees in the angle is N. Here N could be a proper or improper fraction or a real number.
Now , the two measures of the central angle are proportional. So for the same central angle:
a radians = K( N degrees)
Now 2 p radians = K (360 degrees) or p radians = K (180 degrees) implies the proportionality constant
| K | = | p radians
180 degrees |
Therefore the measure in radians is
| a radians | = | p
radians
180 |
N |
= | p
N
180 |
radians |
or the number a of radians is
| a | = | p
180 |
N |
where N is the number of degrees. Conversely, the number N of degrees is
| N = | 180
p |
a | . |
where a is the number of radians.
For the sake of precision in algebraic reasoning, the ratio [(p)/180] and its reciprocal [180/(p)] should carried through calculations exactly and only replaced by their approximations when actual computations are required. The earlier replacement is imprecise and making may cause opportunities for cancellation of the terms p and 180 to be missed.
Remark: For a = 1, we obtain
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That is
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Now 360 degrees subtends a full revolution of a circle and 2 p radians. Therefore
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Calculator Usage
When you push buttons to have a calculator compute the value of a sine, cosine or tangent, you set the calculator into degree or radian mode to identify the input as the number N of degrees or the number a of radians. These calculator inputs and outputs are real numbers.
The use of radian measure has no immediate advantage (apparent to the site author) over the use of degree measure for angles. But in higher mathematics, engineering and quantitative sciences, there are sequences of polynomial approximations to trig and exponential functions whose coefficients become simple fractions if angles are measured in radians. Measure in degrees would insert powers of the radian to degree proportionality constant
180
pinto those sequences polynomial approximations, more formally known as Maclaurin and Taylor series. So higher mathematics, engineering and quantitative sciences is simplified by the use of radian measure. Maclaurin and Taylor series may been seen after the discussion of derivative and integration in calculus.
By using (real) number of radians as input, trig functions have real numbers for their domain and range.
Unit in Computations
As shown in the discussion of proportionality at this site, units of measure may be carried algebraically through calculations. Doing so avoids the need to change all unit of measurement in a geometric or physical situation to those in a system of units. Such changes rescale the coefficients or number of units. Carrying units through calculations minimizes and postpones the number of changes of units that may be required. Units may be cancelled or change during the calculation. That being said, units at the end of a calculation have to be correct (what is expected). Here incorrect signal an error in the solution. The possibility of that error signal is another advantage of the carry the units though computation choice. The alternative is to eliminate the units.Simplifying Convention
There is a convention which identify radians, degrees and revolutions with real numbers. The conventions are as follows:
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With this convention, we write radians besides the number a of radians in the radian measure of an angle, and we may carry both radians and degrees units algebraically through computations. Then when computation have to be done, we apply the above conventions.
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Appendix: Angle Measurement Revisited
To measure the angle of the inner arc in degrees, divide he perimeter of the inner circle into 360 equal-length arcs. Then count how many of these inner circle arcs are covered by the inner arc. The length of the inner arc is proportional to the second count. Both counts give the length of the outer and inner arcs in terms of one three hundred and sixtieth (1/360) of the respective perimeters of each their circles In the above diagram, the two arcs, more precisely, arc lengths, s1 and s2 cover the same proportion of the circles of their respective circles. Therefore the two counts must be equal. One or both circles may represent circles could be the perimeter of a protractor. |
ADVERSTISEMENT
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Extras: Not all perfect.
Equal Sign Use/Abuse Real Numbers Say More Positive Linear Inequalities Triangle Inequality Absolute Value |x| |x| Eq'ns & Inequalities Rectangular Coords Shortest Path Distance Formulas Add & Multiply Points Polar Coordinates Radians (A) Vectors (A) Coordinate Arithmetic (A) Navigation on Maps (A) Addition Geometrically (A) Rotation (PT) Translations (PT) Dilatations PT: Rotations
Links to Site Pages outside this site area follow - co- and pre- requisites.
Easy Consequences of this (newest) Complex Number. Starter Lesson follow below to provide an alternate development of HS or college maths.
Vec & Cmplx No Applet
B2 C. Conjugates
B3 Pythagoras
B4 Distance
B5 Rt Triangle Similarity
B6 Trig., Functions
B7 Dot & Cross Products
B8 Cosine Law
B9 Exponential & cis fns
B10 Easy Trig Identities
B11 Set Viewpoint Arithmetic Videos - Real Player Format
Decimal Addition Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division Methods
Fractions
Primes
Greatest Common Divisor
Divisors
Least Common Multiples
Square Root
SimplificationUsing formulas forwards & Backwards - A unifying theme for algebra skill development - the 4th skill in Volume 2, Three Skills for Algebra!