Trigonometric Identities Page 4
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Trigonometric Identities
θ
1 - cos θ
sin θ
tan
=
=
2
sin θ
1 + cos θ
( 20 )
All of the above relationships are easily proved from Euler's identity
i θ
= cos θ + isin θ,
e
( 21a )
and it also follows that
= cos θ - isin θ,
-iθ
e
( 21b )
i θ
-iθ
( 22 )
e
+ e
cos θ =
= cos (-θ)
2
i θ
-iθ
e
- e
sin θ =
= - sin (-θ)
( 23 )
2i
and these identities can be manipulated to get a new and sometimes more
convenient expression for the trigonometric function of an angle. Just in case
you doubt this method, we append some derivations:
2
2
iθ
-iθ
iθ
-iθ
e
+ e
e
- e
2
2
θ + sin
θ =
cos
+
2
2i
2iθ
-2iθ
2iθ
-2iθ
e
+ 2 + e
e
- 2 + e
=
+
4
-4
( 24 )
2iθ
-2iθ
2iθ
-2iθ
e
+ 2 + e
- e
+ 2 - e
=
4
= 1 .
2iθ
-2iθ
e
- e
sin 2θ =
2i
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