Trig Identities Simplified
The verification or derivation of trig identities can be reduced to algebraic
manipulations involving the cis(q) function
| cis(q) =
cos(q) + i sin(q)
= exp(i q) |
|
and the property
| exp(ia)
exp(ib) = exp(i(a+b)) |
|
The latter property, as said, follows from the polar coordinate, add the
angle, multiply the lengths rule for multiplication of complex
numbers.
The demonstration of many trig identities without the use of the properties
of the function cis(iq) is an enormous task,
or a delicate procedure, a needless but traditional exercise in some
trigonometric courses.
BabyTalk: Trig students meet the cis function (cosine i sine
function) when teachers or course design prefer not to speak about
exponentials exp(i q) of imaginary or
complex numbers.
Angle Sum Formulas, Two Proofs
Two proofs of the angle sum formulas are given next. The first recalls the
rotate-a-triangle proof met in trigonometry. The second derives the formulas in
three steps from the add the angles, multiple the lengths definition of
complex multiplication. Both proofs rely on the addition of angles and rotation
of points on the unit circle. The rest of this section is optional reading
especially if you have seen proofs of these angle sum formulas.
First Proof
This proof of the angle sum formula for sines and cosines involves the
rotation about the origin by the angle b of a
triangle formed by the vertices (0,0), (1,0) and (cos(a-b),sin(a-b))
into the triangle with vertices (0,0), (cos(b),sin(b))
and (cos(a),sin(a))
respectively. We assume this rotation does not change the lengths of the sides
of the triangle - that it is, a rigid body motion.
The original triangle and the rotated triangle both have a side that is a
chord on the unit circle. The length of this side is not changed by the
rotation, at least that is our geometric assumption. Therefore computation of
chords length squared can be done with the help of its end-point coordinates
before and after rotation. And the two computations should give the same result.
This implies
cos(a-b)-1)2+(sin(a-b)-0)2
= (cos(a)-cos(b))2+(sin(a)-sin(b))2
The latter in turn implies
| 1-2cos(a-b)+(cos(a-b))2+(sin(a-b))2 |
|
|
|
|
| (cos(a))2-2cos(a)cos(b)+(cos(b))2 |
|
|
|
|
| +(sin(a))2-2sin(a)sin(b)+(sin(b))2 |
|
|
Therefore
|
|
|
| 2-2cos(a)cos(b)
-2sin(a)sin(b) |
|
|
From this,
| cos(a-b)
= cos(a)cos(b)
+sin(a)sin(b) |
|
The replacement of b by -g
yields
| cos(a+g)
= cos(a)cos(g)-sin(a)sin(g) |
|
as sin(-g) = -sin(g).
The identity cos(q) = sin(90°-q)
implies the identity cos(90°-q)
= sin(q). Together, the identities imply
|
|
|
|
|
|
|
| cos(90°-a)cos(-g)-
sin(90°-a)sin(-g) |
|
|
|
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| sin(a)cos(-g)-
cosa)(-sin(g)) |
|
|
|
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| sin(a)cos(g)+
cosa)(sin(g)) |
|
|
as cos(-q) = cos(q). This
completes the rotate-a-triangle proof for the angle-sum formulas.
| |
Why Slopes
and
More Math
understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
|
[ Back ] [ Home ] [ Next ]
Content Guide
Foreword
Chapter Descriptions
1. Introduction
Preview -why slopes in 1983
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review (optional)
9 On Calculus Studies
11 Slope of Slope
13 Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17 What is Area
18 Integration
18 Area Calculation
18 Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity Motions
10 Slopes without Units.
10 Units & Slopes
10 Units in Cost vs. Quantity
10 How Units Appear
10 Unit Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
Theorem
Integration
& Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.
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