Foreword to Trigonometry
Methods for Solving Geometric Problems
In the first instance, trigonometry is about finding missing sides and
angles in triangles using six ratios or functions: sine, cosine,
tangent, secant, cosecant and cotangent. In that trigonometry with
function values given by tables, electronic calculators or ratio of sides
of two special triangles gives alternate paths for finding mising lengths
and angles.
Before use of trigonometry, we may drawn carefully diagrams to scale
using given information, and then by measurement and/or scaling obtain
the missing sides and angles. The accuracy of results depends on the care
and precision of the triangles. Using diagrams, maps and plans drawn to
scale works when triangle construction methods do not fail or are not
ambiguous. Failure or ambiguity arises when the data given to describe or
imply a triangle implies none or more than one fits the data. Before
electronic geopositioning instruments, ship and aircraft navigators would
(I presume) use maps or charts drawn to scale to plot and plan ship and
aircraft routes. Along the way, they would look for landmarks or take
celestial sightings to check and correct their estimates of their
location and course.
The use of maps, plans, diagrams and charts drawn to the same scale in
all directions, so that angles and shapes are not distorted, provides an
alternative to the use of trigonometry. That being said, the use of
trignometry allows maps, plans, diagrams and charts to be drawn
approximately because the latter will not be used for measurement of
angles and scaled lengths. Diagrams and coordinates instead will be used
in calculations,calculations which require accurate or exact numbers but
diagrams drawn to scale. The diagrams simply require drawing to provide
picture of the situation at hand.
In junior high school mathematics, one student told me that explanations
of why methods or formulas work in mathematics is not necessary. As an
instructor, the school employed me to give correct formulas. Those
formulas being correct did not need to be justified. She was partially
correct. In mathematics with take home value, learning how to use
mathematical methods is important, otherwise their take-home value will
be missed. But the study of geometry and trigonometry in senior high
school mathematics is not for its take-home value. The study is
preparation (wanted or not) for college programs in science, technology
and business. The latter are mathematical disciplines where the skilful
use of mathematics in practice, theory or both, requires comprehension of
the origins and limitations of methods - why they work and where they may
fail or become ambiguous. While there is an initial emphasis on using
trigonometry to find missing angles and lengths in isolated triangles and
then in overlapping ones, most of the missing angles and lengths could
have been found by drawing diagrams to scale. Solving triangles using
trigonometry is essentially an artificial exercise in the high school
level preparation for calculus - the key to college programs in
mathematical disciplines.
Methods for Solving Geometric Problems
-
Mostly Graphical. On a diagram, map or plan. Draw the
geometric information to scale. Then measure missing or needed angles
as is and missing or needed lengths. Scale factors will be needed to
convert drawn lengths and areas into actual lengths and areas. This
method is graphical except for the use of scale factors, one for
length and another for areas. Shapes or regions in question do not
have to be triangular but could be.
-
Using Proportionality or Similarity Relations.In the case of
missing lengths, if two shapes are known to be similar then lengths
and areas [the dimensions] in one are proportional to lengths and
areas in another. Similarity is here described by algebraic
proportionality relations or equations with one or more scale
factors. Those equations may be employed forwards and backwards to
find missing scale factors, and then missing dimension related
measures. Shapes or regions in question do not have to be triangular
but could be.
-
Use the Pythagorean Formula. In the case of right triangles,
the lengths of the hypotenuse and the legs - the two other sides -
are related by the Pythagorean equation. This equations may be
employed forward and backwards to obtain missing lengths.
-
Use sines, cosines and tangent ratios.
\begin{eqnarray*}
\sin(A)&\&frac{opposite}{\mbox{hypotenuse}} \\
\cos(A)&\&frac{adjacent}{\mbox{hypotenuse}} \\
\tan(A)&\&frac{opposite}{\mbox{adjacent}} \\
\end{equarray*}
forwards and backwards.
Here missing angles and
lengths may be found by employing these three trig ratios forwards
and backwards. Shapes here must be right triangles or composed of
them in an overlapping manner. Two angles in the right triangle will
be acute
-
Use the sine and cosine law. These two laws or equations
provide relations between the sine and cosines of angles in a
triangle, not necessarily a triangle, and the lengths of its sides.
The cosine law includes the Pythagorean theorem and its converse as a
special case or cases. These triangles do not have to include a right
angle. For full comprehension and development, how to calculatef
sines and cosines for acute, right and obtuse angles needs to be
explained. That being said, some mathematics course take a shortcut of
telling students to press buttons on a calculator. That would be fine
except that trigonometry studies are nominally part of the mathematical
preparation for calculus.
Use Distance Formulas. The distance between points along
a number line is given by the absolute value of their difference. The distance
between points in a plane or space in the presence of coordinates may be
given by a Pythagorean-Based distant formulas. All can be used forwards
and backwards.
Solving right triangles with the aid of ratios for
sine, cosine and tangent, and solving other triangles with the aid of cosine
and sine laws, most represent artifical exercises to provide familarity
with trigonometry first for acute and then for further angles.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
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How Texas sent
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May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
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Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
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gives boys and girls a head start. Good luck. At the other
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McCainian: drill, drill, drill then Toronto
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Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
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Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
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Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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