36
Chapter 1
•
Right Triangle Trigonometry
§1.5
Notice that reflection around the
y
axis is equivalent to reflection around the
x
axis (
θ
mapstochar→
−
θ
) followed by a rotation of 180
◦
(
−
θ
mapstochar→−
θ
+
180
◦
=
180
◦
−
θ
), as in Figure 1.5.7.
x
y
θ
−
θ
r
r
(
x
,
y
)
(
x
,
−
y
)
(
−
x
,
y
)
180
◦
−
θ
r
Figure 1.5.7
Reflection of
θ
around the
y
axis
=
180
◦
−
θ
It may seem that these geometrical operations and formulas are not necessary for evalu
ating the trigonometric functions, since we could just use a calculator. However, there are
two reasons for why they are useful. First, the formulas work for any angles, so they are
often used to prove general formulas in mathematics and other fields, as we will see later
in the text. Second, they can help in determining which angles have a given trigonometric
function value.
Example1.27
Find all angles 0
◦
≤
θ
<
360
◦
such that sin
θ
=−
0
.
682.
Solution:
Using the
✄
✂
a0
✁
sin
−
1
button on a calculator with
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 Fall '11
 Dr.Cheun
 Calculus, Trigonometry, Law Of Cosines, Inverse function, Leonhard Euler, Euler's formula

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