Home page for accesible maths
Math 101 Chapter 4: Taylor series and complex numbers
4.40 Simplifying multiple angles in terms of trig powers
4.42 Expressing trig powers in terms of multiple angles
Style control - access keys in brackets
Font (2 3)
-
+
Letter spacing (4 5)
-
+
Word spacing (6 7)
-
+
Line spacing (8 9)
-
+
4.41 Trig functions in terms of
z
z
Trigonometric functions in terms of complex exponentials
(i) Let
z
=
e
i
θ
z=e^{i\theta}
. Then
2
cos
θ
=
z
+
1
z
,
2
i
sin
θ
=
z
-
1
z
.
2\cos\theta=z+{{1}\over{z}},\qquad 2i\sin\theta=z-{{1}\over{z}}.
This is equivalent to
2
cos
θ
=
e
i
θ
+
e
-
i
θ
,
2
i
sin
θ
=
e
i
θ
-
e
-
i
θ
.
2\cos\theta=e^{i\theta}+e^{-i\theta},\qquad 2i\sin\theta=e^{i\theta}-e^{-i% \theta}.