TRIGONOMETRIC IDENTITIES - Vibrationdata
COMPLEX FUNCTIONS AND TRIGONOMETRIC IDENTITIES
Revision E
By Tom Irvine
Email: tomirvine@
September 14, 2006
Trigonometric Functions of Angle (
[pic]
(1)
Trigonometric Expansion
[pic] (2)
[pic] (3)
[pic] (4)
[pic] (5)
Exponential Expansion
[pic] (6)
Trigonometric Identities
[pic] (7)
[pic] (8)
[pic] (9)
[pic] (10)
[pic] (11)
[pic] (12)
[pic] (13)
[pic] (14)
[pic] (15)
[pic] (16)
[pic] (17)
[pic] (18)
[pic] (19)
[pic] (20)
[pic]
(21)
Euler's Equation
[pic] (22)
[pic] (23)
[pic] (24)
Hyperbolic Functions
[pic] (25a)
[pic] (25b)
[pic] (26)
Derivatives
[pic] (27)
[pic] (28)
[pic] (29)
[pic] (30)
[pic] (31)
[pic] (32)
[pic] (33)
[pic] (34)
[pic] (35)
Natural Logarithm of a Complex Number
[pic] (36)
[pic] (37)
[pic] (38)
[pic] (39)
APPENDIX A
The Square Root of a Complex Number.
Consider
[pic] (A-1)
Thus
[pic] (A-2)
where a and b are real coefficients.
Solve for x.
Let
[pic] (A-3a)
[pic] (A-3b)
where c and d are real coefficients.
Substitute equation (A-3a) into (A-1).
[pic] (A-4)
[pic] (A-5)
[pic] (A-6)
Equation (A-6) implies two equations. The first is
[pic] (A-7)
The second implied equation is
[pic] (A-8)
Solve for d using equation (A-8).
[pic] (A-9)
Substitute equation (A-9) into (A-8).
[pic] (A-10)
[pic] (A-11)
Multiply through by [pic]
[pic] (A-12)
Apply the quadratic formula.
[pic] (A-13)
[pic] (A-14)
[pic] (A-15)
[pic] (A-16)
Require c to be real.
[pic] (A-17)
Substitute equation (A-17) into (A-9).
[pic]
(A-18)
[pic]
(A-19)
[pic]
(A-20)
Substitute equations (A-20) and (A-17) into (A-3a).
[pic] (A-21a)
Substitute equations (A-20) and (A-17) into (A-3b).
[pic] (A-21b)
Note that equations (A-21a) and (A-21b) cannot be used for the special case:
a < 0 and b = 0.
For this special case, the roots are
[pic] (A-21c)
Example
[pic] (A-22)
[pic] (A-23)
Solve for x. Use equation (A-21a).
[pic] (A-24)
[pic] (A-25)
[pic] (A-26)
[pic] (A-27)
APPENDIX B
Arbitrary Root of a Complex Number
Let
[pic] (B-1a)
[pic] (B-1b)
The coefficients a and b are real numbers. The denominator of the exponent n is also real.
Take the natural logarithm.
[pic] (B-2)
[pic] (B-3)
[pic] (B-4)
[pic] (B-5)
[pic] (B-6)
[pic] (B-7)
[pic] (B-8)
[pic] (B-9)
[pic] (B-10)
Note that equation (B-10) could be used for the special case of a square root.
APPENDIX C
Cube Root of a Complex Number
Consider
[pic] (C-1)
[pic] (C-2)
Equation (C-1) has three roots. The method in Appendix B yields the following formula for one of the cube roots.
[pic] (C-3)
Rearrange equation (C-1).
[pic] (C-4)
Devise an equation for finding the other two roots.
[pic] (C-5)
Expand the right-hand-side.
[pic] (C-6)
[pic] (C-7)
[pic] (C-8)
[pic] (C-9)
[pic] (C-10)
Equation (C-10) implies three separate equations.
[pic] (C-11)
[pic] (C-12)
[pic] (C-13)
Continue with equation (C-11).
[pic] (C-14)
Substitute equation (C-14) into (C-12).
[pic] (C-15)
[pic] (C-16)
[pic] (C-17)
[pic] (C-18)
Use the quadratic formula.
[pic] (C-19)
[pic] (C-20)
[pic] (C-21)
[pic] (C-22)
Choose
[pic] (C-23)
Recall equation (C-14).
[pic] (C-24)
[pic] (C-25)
[pic] (C-26)
[pic] (C-27)
[pic] (C-28)
[pic] (C-29)
The roots x2 and x3 thus form a complex conjugate pair.
Summarize the roots.
[pic] (C-30)
[pic] (C-31)
[pic] (C-32)
Example
Solve for x.
[pic] (C-33)
[pic] (C-34)
[pic] (C-35)
[pic] (C-36)
n = 3 (C-37)
There are three roots. The first root is
[pic] (C-38)
[pic] (C-39)
[pic] (C-40)
The second root is a coordinate transformation of the first root.
[pic] (C-41)
[pic] (C-42)
[pic] (C-43)
[pic] (C-44)
[pic] (C-45)
[pic] (C-46)
In summary, the cube roots of (2 + j 7) are
[pic] (C-47)
[pic] (C-48)
[pic] (C-49)
APPENDIX D
Derivation of the Quadratic Formula
[pic] (D-1)
[pic] (D-2)
[pic] (D-3)
[pic] (D-4)
[pic] (D-5)
[pic][pic] (D-6)
[pic] (D-7)
[pic] (D-8)
[pic] (D-9)
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