Review of Trigonometry, Vector Algebra and Calculus
1.
Review of Trigonometry
1.1.
Angles
An
angle
is defined as the portion of a plane between two lines. An angle that encom
passes the entire plane is said to measure 2
π
radians, or 360 degrees.
C
a (adjacent side)
(angle)
α
o (opposite side)
Fig. 1.—
Definition of angle, and adjacent and opposite sides to an angle .
An angle is measured by taking the ratio of the
arc
subtended by the angle, and the
radius of any circle centered where the two lines meet (point C). The unit of measure of
angle is the
radians
.
This is a peculiar unit of measure, as it is
dimensionless
, since it is
the ratio of two lenghts. Radians can be converted to
degrees
remembering that 2
π
radians
correspond to 360 degrees:
α
[deg] =
360
2
π
×
α
[rad]
(1)
1.2.
Trigonometric functions
Consider the right triangle formed by the intersection of the two lines with any circum
eference centered where the two lines meet (Figure 1). The adjacent and opposite sides, and
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– 2 –
the radius of the circle, define the following trigonometric functions:
1.
sine
:
sin
(
α
)
≡
o
r
2.
cosine
:
cos
(
α
)
≡
a
r
3.
tangent
:
tan
(
α
)
≡
sin
(
α
)
cos
(
α
)
=
o
a
4.
cotangent
:
cotan
(
α
)
≡
1
tan
(
α
)
The sine and cosine functions have a periodic behavior, which repeats itself every 2
π
radians:
a (angle)
π
2π
cosine
sine
+1
1
Fig. 2.—
Sine and cosine functions.
There are few important trigonometric relationship that it is useful to remember:
•
sin
2
(
α
) +
cos
2
(
α
) = 1
•
cos
(
α
±
β
) =
cos
(
α
)
·
cos
(
β
)
∓
sin
(
α
)
·
sin
(
β
)
•
sin
(
α
±
β
) =
sin
(
α
)
·
cos
(
β
)
±
cos
(
α
)
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 Fall '08
 BONAMENTE
 Physics, Trigonometry, Mathematical analysis, Tn, dx, Chattanooga

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