Trigonometry Worksheet Page 14
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22%
%
Since tan
=
, we conclude that tan
=
.
%
%
Right-Triangle Trigonometry and the Reciprocal Functions
The sine, cosine and tangent functions can be defined on a right triangle. Let angle
be placed as
shown in Fig. 11.
Fig. 11
Remember, the leg closest to this angle is called the adjacent leg, and the other leg is called the opposite
leg. Therefore, the sine, cosine and tangent functions can be defined in terms of the lengths of the
adjacent leg, the opposite leg and the hypotenuse:
GCH?
>D =? H
GCH?
sin =
, cos =
, tan =
.
BM
H? IG?
BM
H? IG?
>D =? H
Using “O” for opposite, “A” for adjacent and “H” for hypotenuse, the mnemonic “SOHCAHTOA” is
a useful way to commit these relationships to memory. Often, the trigonometric values can be
determined even if the angle is unknown, as the following example shows.
Example: A right triangle as an adjacent leg of length 7 and a hypotenuse of length 10. Determine exact
values for sin , cos and tan .
Solution: We use the Pythagorean Formula to determine the length of the opposite leg (denoted y):
7
+
= 10
$
$
$
49 +
= 100
$
= 51
$
= 51
Therefore,
'#
>D
'#
sin =
=
, cos =
=
and tan =
=
.
BM
#"
BM
#"
>D
The reciprocal functions are defined as follows:
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