Trigonometry Worksheet Page 9
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22$
$
Using radians, we have cos
=
, sin
=
and tan
= 1.
&
$
&
$
&
Now suppose we draw the ray at an angle of 60°
radians , intersecting the unit circle at point P.
%
Recall from geometry that an equilateral triangle (all sides equal in length) has internal angles of 60°.
Therefore, we draw a line from P to the point (1,0) and form an equilateral triangle with sides of length
#
, 0 (See Fig 10).
1. Now divide this triangle in half by sketching a vertical line from P to the point
$
Fig. 10
#
We now have a right triangle with hypotenuse of length 1 and an adjacent leg of length
. Using the
$
%
#
%
,
Pythagorean formula, the opposite leg has length
. Point P’s coordinate is
. and we have
$
$
$
#
%
cos { 60 { =
and sin{60{ =
.
$
$
The slope of this ray is
GC { "{
% $
tan{60{ =
=
= 3.
= G { "{
#/$
#
%
In radians, we have cos
=
, sin
=
and tan
= 3.
%
$
%
$
%
%
#
%
and tan { 30 { =
A similar construction shows that cos{30{ =
, sin{30{ =
. In radians, we have
$
$
%
%
#
%
cos
=
, sin
=
and tan
=
. This is left as homework exercise XX.
$
$
%
These values are summarized in the following table:
cos
sin
tan
Angle
Angle
(Degrees)
(Radians)
0
0
1
0
0
30°
6
3 2
3 3
1/2
45°
4
2 2
2 2
1
60°
3
3 2
3
1/2
90°
2
undefined
0
1
Note that these values are for the first quadrant.
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