64 Chapter 3Transcendental functions
Several derived functions are also defined, the most important being the secant(sec),
cosecant(cosec), and cotangent (cot):
(3.4)
EXAMPLE 3.1For the angles in Figure 3.3,
0 Exercise 1
One of the best known properties of the right-angled triangle is the theorem of
Pythagoras.
2AB
21 + 1 BC
21 = 1 AC
2(3.5)
This can be written as a trigonometric equation by dividing both sides by AC
2:
or
sin
21 θ+cos
21 θ 1 = 11 (3.6)
(a quantity like(sinθ)
2, the square ofsin 1 θ, is usually written assin
21 θ).
AB
AC
BC
AC
2222+= 1
cosecφφφ===
5
3
5
4
4
3
sec cot
sinφφφ===cos tan
3
5
4
5
3
4
cotθ=
3
4
secθ=
5
3
cosecθ=
5
4
tanθ=
4
3
cosθ=
3
5
sinθ=
4
5
sec
cos sin
cot
tan
cos
sin
θ
θ
θ
θ
θ
θ
θ
θ
=, =, ==
111
cosec
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.......................................A B
C
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..............θ
φ
4
5
3
Figure 3.3
2Pythagoras (c.580–c.500 BC). Born on the island of Samos, he travelled widely and was one of the principal
importers of Egyptian and Babylonian mathematics and astronomy into the Greek world. He settled in Croton, in
Southern Italy, where he founded a religious and philosophical society with a strong mathematical basis; motto
‘all is number’. He is reputed to have coined the word ‘mathematics’; that which is learned. Attributions to him
of mathematical discoveries are traditional; Pythagoras’ theorem was known in the old Babylonian period, and
the existence of irrational numbers was possibly a discovery of the later Pythagoreans in about 400 BC. The
Pythagorean school introduced the systematic study of the principles of mathematics; number theory and geometry.
The general form of the theorem of Pythagoras is: ‘In a right-angled triangle, the area of the figure on the
hypotenuse is equal to the sum of the areas of the similar figures on the other two sides’ (Euclid, ‘The elements’,
Book VI, Proposition 31 in the Heath translation).