Graphics Reference
In-Depth Information
hypotenuse
opposite
b
adjacent
Fig. 4.1.
Labeling a right-angle triangle for the trigonometric ratios.
4.1 The Trigonometric Ratios
Ancient civilizations knew that triangles, whatever their size, possessed some
inherent properties, especially the ratios of sides and their associated angles.
This meant that if such ratios were known in advance, problems involving
triangles with unknown lengths and angles could be computed using these
ratios.
To give you some idea why we employ the current notation, consider the
history of the word sine. The Hindu word
ardhajya
meaning 'half-chord' was
abbreviated to
jya
('chord'), which was translated by the Arabs into
jiba
,
and corrupted to
jb
. Other translators converted this to
jaib
, meaning 'cove',
'bulge' or 'bay', which in Latin is
sinus
.
Today, the trigonometric ratios are commonly known by the abbreviations
sin, cos, tan, cosec, sec and cot. Figure 4.1 shows a right-angled triangle where
the trigonometric ratios are given by
opposite
hypotenuse
adjacent
hypotenuse
tan(
β
)=
opposite
adjacent
sin(
β
)=
cos(
β
)=
1
sin(
β
)
1
cos(
β
)
1
tan(
β
)
cosec(
β
)=
sec(
β
)=
cot(
β
)=
The sin and cos functions have limits
±
1, whereas tan has limits
±∞
.The
signs of the functions in the four quadrants are
+ +
− −
−
+
−
+
−
+
+
−
sin
cos
tan
4.2 Example
Figure 4.2 shows a triangle where the hypotenuse and one angle are known.
The other sides are calculated as follows:
h
10
= sin(50
◦
)
