Trigonometry
If you took a course in trigonometry, and spent hours on the many esoteric minutiae
that it can lead to, I want to assure you that the number of those concepts that you use
in daytoday work is small. But, you use
those
every day: The deﬁnitions of the sine,
cosine, and tangent. A few of the identities among them. Mostly, how to recognize them in
unfamiliar contexts.
The basis of trigonometry lies in the fact that for similar triangles, the ratio of cor
responding sides is the same. Remember that similar triangles have the same shape (not
necessarily the same size), so that their
angles
are equal. The trigonometric names are simply
the titles given to these ratios in the particular case of a right triangle.
a'
c
b
θ
a
b'
c'
The deﬁnitions are
sin(
θ
) =
a
c
=
a
0
c
0
,
cos(
θ
) =
b
c
=
b
0
c
0
,
tan(
θ
) =
a
b
=
a
0
b
0
.
The Greek letter
θ
(theta) is commonly used for angles, as is the Greek letter
φ
(phi). The
other deﬁnitions, such as the cotangent: cot
θ
= 1
/
tan
θ
show up too, but less often.
From the deﬁnitions, you immediately get an identity:
tan
θ
=
a
b
=
a/c
b/c
=
sin
θ
cos
θ
.
Similarly, start from the Pythagorean theorem,
a
2
+
b
2
=
c
2
, and divide both sides by
c
2
:
a
2
+
b
2
=
c
2
=
⇒
a
2
c
2
+
b
2
c
2
= 1
,
or
sin
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 Fall '11
 Galeazzi
 Pythagorean Theorem, Law Of Cosines, Work, triangle, 0.500 M, 1.87 M, 1.00 meters

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