TRIGONOMETRIC IDENTITIES



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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