|
YOU are better than YOU think. Show yourself
how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
Logic
chapters 1 to 5 re- appear not in sequence, as is or longer, in
Volume 1A, Pattern Based Reason,
Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
| | Derivatives of sine and cosine
Differentiation Rules for sine and cosine
- sin'(x) = cos(x) and
- cos'(x) = -sin(x).
|
The following sections explain why the derivative of cos(x) is -sin(x) and
why the derivative of sin(x) is cos(x).
From Earlier Pages, recal
|
lim
h® 0 |
cos(h) - 1
h |
= |
0 |
|
|
|
|
|
lim
h® 0 |
sin(h)
h |
= |
1 |
|
|
|
|
|
6 A. Derivatives of sin(x) and cos(x) - real number way
A complex number way follows below.
Recall the angle sum formulas
cos(a+b) = cos(a)cos(b)
- sin(a)sin(b)
sin(a+b) = sin(a)cos(b)
+ cos(a)sin(b)
Now
cos(x+h) = cos(x)cos(h)- sin(x)sin(h)
gives
|
lim
h® 0 |
cos(x+h) -cos(x)
h |
= |
lim
h® 0 |
(cos(x)-1)cos(h)- sin(x)sin(h)
h |
|
|
|
= |
lim
h® 0 |
(cos(x)-1)
h |
cos(x) - sin(x) |
sin(h)
h |
|
|
|
|
|
= |
|
0 |
cos(x) - sin(x) |
1 |
|
|
|
|
= |
- |
sin(x) |
|
|
|
|
Therefore cos'(x) = -sin(x).
Similarly,
sin(x+h) = sin(x)cos(h) + cos(x)sin(h)
gives
|
lim
h® 0 |
sin(x+h) -sin(x)
h |
= |
lim
h® 0 |
sin(x) (cos(x)-1)+
cos(x)sin(h)
h |
|
|
|
= |
lim
h® 0 ) |
(cos(x)-1)
h |
sin(x) + cos(x) |
sin(h)
h |
|
|
|
|
|
= |
|
0 |
sin(x) + cos(x) |
1 |
|
|
|
|
= |
|
cos(x) |
|
|
|
|
Therefore sin'(x) = cos(x).
Derivatives of Basic Trig Functions
Now the derivatives of the other four basic trig functions tan(x),
cot(x), sec(x) and csc(x) follow from the quotient and reciprocals rules for
differentiation along with some simplification based on the definition of trig
functions in terms of sines and cosines, and the Pythagorean identity.
6B. Derivatives of sin(x) and cos(x) - complex number way (optional)
| If you knew about complex
numbers and
cis(x) = cos(x) + i sin(x)
we could at this point calculate the derivative of cis(x),
and its real and imaginary parts cos(x) and sin(x) as follows.
Note if f(x) = h(x) + ig(x) for a pair of real
valued function of pair of real-valued variable x, we take
lim x® a f(x) = lim x® a h(x) + i lim x® a g(x)
These complex-valued limits inherit many of the
properties of the real-valued limits studied earlier. Proofs are
omitted. It is left to the student (or another course) to formally state
the properties.
It is easy to show lim x® a f(x) k = [lim x® a f(x) ] k for any complex number k that is independent of
x.
By our definition of complex-valued limits (saying how to
compute them, defines them), we have
|
lim h® 0 |
cis(h)- 1
h |
= |
lim h® 0 |
(cos(h) -1) + i sin(h)
h |
|
|
|
|
|
= |
lim h® 0 |
(cos(h) -1)
h |
+ i |
sin(h)
h |
|
|
= |
|
0 |
+ i |
1 |
|
|
= |
i |
|
|
|
Now
|
lim h® 0 |
cis(x+h)- cis(x)
h |
= |
lim h® 0 |
(cis(h)- 1)cis(x)
h |
|
|
|
|
= |
i |
cis(x) |
|
|
That is to say
cis'(x) = i cis(x) = i [cos(x) + i sin(x)] = -
sin(x) + i sin(x).
But cis'(x) = cos'(x) + i sin'(x) as well. The argument
that cis'(x) implies cos'(x) and sin'(x) exist as well, and both are real.
So
cis'(x) = cos'(x) + i sin'(x) = - sin(x) + i sin(x).
and comparison of real and imaginary (equating them) gives
cos'(x) = -sin(x) and sin'(x) = cos(x) |
| |
[ Back ] [ Up ] [ Next ]
More Calculus
Calculus Videos
0. First Calculus Preview
0. Triangle Inequality
0 Inequalities
0. Solving Inequalities
1. Distance+Midpt Formulas
1. Function Domains
1.Polynomial Domain+Range
1. Fn: Linear Combinations
1. Fn Composition II
1. Fn Composition I
1. Solving y**n = x**m
2 .Real Numbers
2. Limits Numerical View
2. Limits,. Formal Definition
2. Limit Properties Numerically
2. Decimal View of Limits
2. Error Control View
2. Limits & Continuity
2. Limit Vals via Substitution
2. Limits & Composite Fns
2.. Limit Examples
2. One Sided Limits
2. Infinity and Limits
2. Parameters in Limits
3. Derivative Motivation
3. Derivative Definition I
3. Derivatives Definition II
3. Calculus: Why Radians
3 d/dx of sin(x) & cos(x) (I)
3 d/dx of sin(x) & cos(x) (II)
3.Sum Rule
3. Product Rule
3. Power Rule
3. Previous Rules Combined
3. d/dx for Polynomials
3. Reciprocal Rule
3. Reciprocal Law: sec & csc
3. Reciprocals & Power Rule
3. Power Law for Integers < 0
3. Quotient Rule
3. Quotient Rule Examples
3. Quotient Rule: tan & cot
3. Linear Chain Rule
3. Chain Rule for Powers
3. Chain Rule - Polynomials
3. Chain Rule Examples I
3. Chain Rule Examples II
3. Linear Approximation I
3. General Chain Rule
3. Inverse Fns Derivatives
3. Chain Rule: ln(x) & exp(x)
3. Square & Cube Roots
4. Linear Approximation
4. Second Derivative Test
4. Sketch y = x^3 - 6x^2- 12x
4. Sketch y = x^3 - 3 x^2 - 9x
4. Sketch y = 1 - 1/(1+x^2)
5. Indefinite Integrals A
5. Indefinite Integrals B.
5 Indefinite Integrals C
6. Definite Integral D
6. Area Under Curves
7. Volume of a Sphere
To Learn More, visit Volumes 2 and 3.
Advanced Topics
Limit Properties Algebraically
Pigeon Hole Principle
Bolzano
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
From Lipschitz Continuity
To Learn More: Visit Real Analysis.
[ Back ] [ Up ] [ Next ]
|