Trigonometric Identities Page 5
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6Trigonometric Identities
5
2
2
iθ
-iθ
e
- e
=
2i
iθ
-iθ
iθ
-iθ
(e
+ e
)(e
- e
)
=
2i
( 25 )
iθ
-iθ
) sin θ
= (e
+ e
= 2 cos θ sin θ .
The hyperbolic functions are analogous to the trig functions and often arise in
physical situations. Their relations to the trig functions are as follows:
x
-x
-i(ix)
i(ix)
e
- e
e
- e
sinh x =
=
= -i sin (ix) ,
( 26 )
2
2
x
-x
-i(ix
i(ix)
e
+ e
e
) + e
cosh x =
=
= cos (ix) ,
( 27 )
2
2
x
cos x = cosh
= cosh (-ix) = cosh (ix) ,
( 28 )
i
x
sin x = i sinh
i = i sinh (-ix) = -i sinh (ix).
( 29 )
(The following laws apply for all
B
a
triangles with angles, A, B and C and
c
opposite side lengths as defined in
the figure.)
A
C
b
sin A
sin B
sin C
Law of Sines:
=
=
( 30 )
a
b
c
2
2
2
Law of Cosines: c
= a
+ b
- 2abcos C
( 31 )
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