Download Trigonometry The word trigonometry literally means “measure of

Trigonometry
The word trigonometry literally means “measure of
triangles.” Its fundamental objects are the
trigonometric ratios for right triangles: If ΔABC
has right angle at C, and as is customary, we label
the angles of the triangle by the names of their
vertices and the sides by a = BC,b = CA, c = AB,
€
then
a
sin A = € is the sine,
c
b
cos A = is the cosine, and
c
a
tan A = is the tangent
b
of (acute) angle A. We can also define cscA (the
cosecant), secA (the secant), and cotA (the
€
cotangent) of A to be the corresponding reciprocals
of the ratios for sine, cosine and tangent. To extend
these definitions to angles other than acute angles,
define
sin A = sin(180 – A) and
cos A = –cos(180 – A)
for any obtuse angle A (definitions for the other trig
functions can be deduced as ratios of these).
Further, to make the sine and cosine continuous
functions, we put sin 0 = 0, sin 90 = 1, sin 180 = 0;
and cos 0 = 1, cos 90 = 0, cos 180 = –1.
Many relationships in trigonometry hinge on the
Theorem [The Pythagorean Theorem] If ΔABC
has right angle at C, then a 2 + b 2 = c 2. //
Corollary [The Pythagorean Identity]
If ΔABC
€
has right angle at€C, then sin 2 A + cos 2 A = 1. //
In fact, the Pythagorean Identity can
€ be used to
give a proof of the
€
Theorem [The Law of Sines] If ΔABC is any
triangle, then
a
b€
c
=
=
. //
sin A sin B sinC
Theorem [The Law of Cosines] If ΔABC is any
2
2
2
triangle,
then
a
+
b
−
2ab
cosC
=
c
. //
€
Corollary [The Cevian Formula]
In ΔABC , if
€
D ∈ A€B , and the segment C D , called a cevian of
the triangle, has length d = CD, which cuts A B
into parts with ratios p = AD/AB
and q = DB/AB,
€
2
2
then pa 2 + qb 2 = pqc
+
d
. //
€
€
€
€
Corollary In ΔABC , if M is the midpoint of A B ,
then the length of the median C M is
€
CM =
1 2
a
2
€
€
+ 12 b 2 − 14 c 2 .€//