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34
Chapter 1
•
Right Triangle Trigonometry
§1.5
Rotating an angle
θ
by 90
◦
in the clockwise direction results in the angle
θ
−
90
◦
. We could
use another geometric argument to derive trigonometric relations involving
θ
−
90
◦
, but it is
easier to use a simple trick: since formulas (1.4)
−
(1.6) hold for
any
angle
θ
, just replace
θ
by
θ
−
90
◦
in each formula. Since (
θ
−
90
◦
)
+
90
◦
=
θ
, this gives us:
sin (
θ
−
90
◦
)
= −
cos
θ
(1.7)
cos (
θ
−
90
◦
)
=
sin
θ
(1.8)
tan (
θ
−
90
◦
)
= −
cot
θ
(1.9)
We now consider rotating an angle
θ
by 180
◦
. Notice from Figure 1.5.4 that the angles
θ
±
180
◦
have the same terminal side, and are in the quadrant opposite
θ
.
x
y
θ
+
180
◦
θ
−
180
◦
(
x
,
y
)
(
−
x
,
−
y
)
θ
180
◦
−
180
◦
r
r
(a)
QI and QIII
x
y
θ
+
180
◦
θ
−
180
◦
(
x
,
y
)
(
−
x
,
−
y
)
θ
180
◦
−
180
◦
r
r
(b)
QII and QIV
Figure 1.5.4
Rotation of
θ
by
±
180
◦
Since (
−
x
,
−
y
) is on the terminal side of
θ
±
180
◦
when (
x
,
y
) is on the terminal side of
θ
,
we get the following relations, which hold for all
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 Fall '11
 Dr.Cheun
 Calculus, Trigonometry

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