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| | Derivatives of sine and cosine
The following sections explain why the derivative of cos(x) is -sin(x) and
why the derivative of sin(x) is cos(x).
Differentiation Rules for sine and cosine
- sin'(x) = cos(x) and
- cos'(x) = -sin(x).
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Suggestion: Leave the explanations below to later. Get an
operational command of differentiation rules and their use first.
1 Angles and ArcLength
(Assume the angle q is measured in degrees)
The length s of a circular arc is proportional to the angle q
its spans. Physically, this proportionality means if the angle q
is double or tripled, so is the arclength s. Mathematically, this
relationship is described by the relation
where k is called the constant of proportionality. (If you graphed s
versus q where 0°
£ q £
360° , you would get a straight line
segment with slope m = k.
The assumption that the circumference of the circle is 2pr
implies 2pr = k·360°
where r is the radius. This in turn implies
The unit of angle measurement is the degree 1°
= 1 degree.
2 Angles and Radian Measure
The foregoing implies that s = kq = [(pr)/(180°
)] ·q = [(p)/(180°
)]rq. This in turn implies
or equivalently that
Thus the real number
is proportional to the angle q, and vice-versa. Thus
the ratio s/r determines the angle q, and
vice-versa. So specifying one, specifies the other. The real number s/r
gives the radian measure of the angle q,Observe r times the radian measure (s/r) gives the arclength s of the
subtended arc. That geometric property provides one motivation for using radian
measure, the ratio of arclength to radius, for angles.
The Calculus Reason Why Radian Measure is important.
When radian measure is used to measure an angle h, the following limit
property
provides the key to differentiation rules for trig functions. The limit
would not have the value 1 if h was measure in degrees. Why the limit
is explained in the next lesson. Bon Appetit.
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More Calculus
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For help in calculus, explore
Volumes
2. Three Skills
for Algebra
and 3. Why
Slopes & More Math, and Calculus
Introduction site area. See how to learn or teach key skills and
concepts, some not all.
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.
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