1.3: The xyr Definitions of the Trigonometric Functions
The six trigonometric functions can be defined in several ways.
The first way may be referred to as the xyr
definitions, where x and y are the coordinates of a point on the terminal side of the angle when placed in standard
position, and r is the distance of that point from the origin.
The inputs to the functions will be angles, and the
outputs will be ratios of the numbers x, y, and z.
Ex
:
Given that
P
(3,5)
=
, compute values
for the six trigonometric functions of the angle
in standard position.
First we compute
r
:
2
2
r
3
5
r
9
25
r
34
=
+
=
+
+
=
We can now write down these ratios, given
that we already know that
x
3
=
and
y
5
=
.
Place
θ
in standard position, pick any point on the terminal side
of
θ
except the origin itself, and use the coordinates of this
point to compute its distance from the origin.
2
2
r
x
y , r
0
=
+
>
=
+
>
+
+
We define the six trigonometric functions as follows.
(sine, cosine, tangent, cotangent, secant, cosecant)
y
sin
r
x
cos
r
y
tan
, x
0
x
x
cot
, y
0
y
r
sec
, x
0
x
r
csc
, y
0
y
θ =
θ =
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 Fall '09
 COHEN
 Trigonometry, Euler's formula, tan θ

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