Page 96 - Numerical Analysis Using MATLAB and Excel

Characteristics of Sinusoids

Let Figure 3.6 be any triangle.

a γ b
β α
c
Figure 3.6. General triangle
Then,
by the law of sines,

a b c
----------- = ----------- = ---------- (3.60)
sin α sin β sin γ
by the law of cosines,
2
2
2
a = b + c – 2bccos α (3.61)
2
2
2
b = a + c – 2accos β (3.62)
2
2
2
c = a + b – 2abcos γ (3.63)
and by the law of tangents,
1 - β – ) 1 - γ – ) 1 )
-- β ( tan
--- γα–( tan
-- α ( tan
ab
2
2
–
2
----------- =
------------ = ------------------------------ b – c ---------------------------- c – a ----------------------------- (3.64)
----------- =
a + b 1 b + c 1 c + a 1
-- α ( tan - β + ) -- β ( tan - γ + ) -- γ ( tan - α + )
2 2 2
The following differential and integral trigonometric and exponential functions, are used exten-
sively in engineering.
dv
d ( sin v = cos v------ (3.65)
)
d x dx
dv
d ( cos v = – sin v------ (3.66)
)
d x dx
d e ( v ) = e ------ (3.67)
v dv
d x dx
∫ sin ax x = – 1 - ax + c (3.68)
--cos
d
a
1
∫ cos ax x = -- sin ax + c (3.69)
-
d
a
∫ e ax d x = 1 ax + c (3.70)
-
--e
a
Numerical Analysis Using MATLAB® and Excel®, Third Edition 3−9
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