Trigonometry or Similarity
a question of convenience
In solving for missing sides or angles in triangles you may be told to
use similarity methods directly or trig functions.
And in answering questions, students should apply the
requested method to show their mastery of it. But if the requested
method has not been mastered, using another method may give part marks
or not.
Trigonometry represents an indirect use of
similarity.
Direct Use of Similarity
Assume we cannot measure the height h directly. To find the height h of
the triangle which casts a shadow 100 metres long when the sun's angle of
elevation in the sky is 50 degrees, we may draw a similar right triangle
to scale with the ASA triangle construction method and base length of 2
centimetres (2 cm)
Now
Therefore
Direct Use of Trig
(Indirect Use of Similarity)
By using trigonometry, we have another way to compute the height H. That
is, we may use the tangent function
to conclude that
H = tan (50o)(100 metres).
Here we may compute tan (50o) with the aid of a table or
calculator. That calculation hides the presence of similarity. But we may
find tan( ) from the right triangle
or any other similar triangle (drawn carefully) using
and our ability measure the opposite side, here h. Then 
Then H = tan (50o)(100 metres) gives
The latter implies
Conclusion or Observation.
In solving problems involving missing sides of a triangle, the use of
tables or calculators to look-up or find the values of a trig function is
but an alternative to computing those values from a triangle similar to
one with the missing sides. Using a calculator or table is more
convenient.
Remark. Before the use of calculator, the values of
trigonometric functions might be tabulated for use in calculations. For
instance, we could by measurement (not a very precise method), estimate
the values of cosine, sine and tangent at 5, 10, 15, 20, ..., 85
degrees by doing measurement and computing the required ratios. The
latter would require many right triangles to be drawn. After that we
could solve for missing sides etc in given right triangles with acute
angles 5, 10, 15, 20, ..., 85 by consulting the table or the triangles
we used to fill the table. We could also interpolate between the values
in the table to obtain estimated values of sine, cosine and tangent for
use in further computations. However, that work has been done for us
(by more refined methods), and the results tabulated in tables and
built-in the computational abilities of calculators.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
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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
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70
Calculus Starter Lessons
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They cover basic topics in ways likely to complement your
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
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if the come easy, may be deceptive - provide a too light and not
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calculus and more generally in the first year of college. Bon
Appetite.
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