Trigonometry Worksheet Page 8
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22as a short-hand for { cos {
This identity is called the Pythagorean Identity. Notice that we write cos
$
$
.
Another very common relationship between sine, cosine and tangent uses the slope formula. Since the
ray passes through the origin (0,0) and point P, which is {cos , sin {, we can calculate the slope of the
ray, which is defined to be tan :
sin − 0
Slope of ray = tan =
cos − 0
Therefore, we have
sin
tan =
cos
Trigonometric Values of Special Angles
Exact values for the sine, cosine and tangent of some special angles can be determined using geometry.
We look at the case of 45°
radians first:
&
Suppose the ray is drawn with an angle of 45° on the unit circle as seen in Figure 9 below. Let P be its
intersection with the circle.
Fig. 9
The hypotenuse has length 1, and its x and y coordinates will be the same. Using the Pythagorean
formula, we have
+
= 1
$
$
Remember, x = y.
2
= 1
$
#
=
$
$
#
$
=
=
.
Rationalizing the
denominator.
$
$
$
$
The x-coordinate of P is
, and the y-coordinate of P is also
. Therefore,
$
$
$
$
cos { 45 { =
and sin { 45 { =
.
$
$
Since the slope is 1, we observe that tan{45{ = 1.
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