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YOU are better than YOU think. Show yourself how: |
-/[]\- Logic
Mastery Do not leave here without it - 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 linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6; 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. |
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Incomplete
12 Trigonometry and a little Similarity
Teachers: The fact that we use the sine and cosine laws for angles between 0 and 180 requires we define trig functions at least in the interval 0 to 180 degrees.
Unit Circle Definition of Trig functions for 0 to 360 degrees with very quick mention or none for angles > 360 or < 0.
Calculation or approximation of trig functions for angles in and possible outside the interval 0 to 360 degrees with the aid of electronic calculators.
The notation convention cos2(q) = (cos(q))2
The Pythagorean Identity
cos2(q) + sin2(q) = 1 for all angles q.
Similarity of Right Triangles
Application: Show how trig functions for acute angles are given by fractions with numerators and denominators equal the length of sided of right triangles. Thus students see the math 436 approach.
Proof and Use of the sine law for acute and obtuse triangles, so angles between 0 and 180 degrees
Proof and Use of the cosine law for acute and obtuse triangles, so angles between 0 and 180 degrees.
Applications of Trig functions and laws alone or with Pythagorean ideas.
Use of trig function definitions, Pythagorean identity, use of sine and cosine laws in solving for missing lengths and angles, one at a time and one after another, in isolated triangles and in entangled (overlapping or adjacent) triangles.
8. Trigonometry
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.
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Lesson & Lesson Plans for
Secondary IV
a reference for learning and teaching functions, polynomials, solving linear
systems,
powers + exponents + bases + radicals (roots) , quadratic formulas, equations
of straight lines
Sit down and study - no one else can do that for you.
Advice and Directions
What to do in School & Why
How
to Study Maths & Why
Good News: If you can learn to follow a multi-step methods in any subject precisely, you should be able to do so in other subjects, as well. Hint: Start with arithmetic
Words Before Symbols:
What is a Variable?
Level: Secondary II to VI, or Grades 7 to 12)
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
Complex number starter lesson
Arithmetic Videos
Fractions
Primes
Greatest Common Divisors
Least Common Multiples
Square Root Simplification
Arithmetic Videos
Decimal Addition Methods
Decimal
Subtraction Methods
Decimal
Multiplication Methods
Decimal Division Methods
Fraction
Starter Lesson
(simplify, multiply, divide &
then add or subtract)